Phương pháp giải toán Hình Giải tích_Dùng LTĐH 2011

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Phương pháp giải toán Hình Giải tích_Dùng LTĐH 2011

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Tài liệu ôn thi đại học môn toán tham khảo gồm hướng dẫn cụ thể về phương pháp giải toán Hình Giải tích, dành cho học sinh hệ trung học phổ thông ôn thi đại học - cao đẳng tham khảo ôn tập củng cố kiến thức.

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  1. sivantran@gmail.com - 01689583116 1. BÀI TOÁN 1: “Lập phương trình đường thẳng đi qua A, vuông góc với đường thẳng (d1) và cắt (d2)” Cách 1: Bước 1: Chuyển phương trình về dạng tham số. uuu r -Giả sử (d) là đường thẳng cần dựng và cắt (d2) tại B, khi đó B(...) ⇒ AB ( ...) . ur ur -Gọi a1 là vtcp của (d1), ta có a1 ( ...) . Bước 2: ur uuu ur r uuu r uuu r Vì (d) (d1) nên : AB a1 ⇔ AB.a1 = 0 (nhớ tích vô hướng) ⇒ AB ( ...) Bước 3: Phương trình đường thẳng (d) được cho bởi:  x = ... qua A ( ...)   ( d ) :  uuu ⇔ ( d ) :  y = ..., t ∈ R. r vtcp AB ( ...)   z = ...  Cách 2: -Giả sử (d) là đường thẳng cần dựng, khi đó (d) chính là giao tuyến của hai mặt phẳng (P1) và (P2), trong đó: qua A ( ...) qua A ( ...) ( P1 ) :   và ( P2 ) :   ( P ) ( d1 )  1 ( d 2 ) ∈ ( P2 )  * Phương trình mặt phẳng (P1) qua A ( ...) qua A ( ...) ( P1 ) :    ⇔ ( P) : r ur ⇔ ( P ) :.... ( P ) ( d1 ) vtptn = a1 ( ...) 1 1  1  * Phương trình mặt phẳng (P2) (mặt phẳng đi qua một điểm và chứa một đường thẳng) Viết phương trình mặt phẳng (P2) bằng 2 cách: Cách 1: Chuyển phương trình (d2) về dạng tổng quát, sau đó sử dụng chùm mặt phẳng. uuuu r Cách 2: Chọn điểm M(...) tùy ý thuộc (d2) ⇒ AM ( ...) qua A ( ...) qua A ( ...)  uu  r uuuu r uu r uuuu uu r r ( P2 ) :   ⇔ ( P2 ) : n2 AM ⇔ n2 =  AM .a2  = ....   ( d 2 ) ∈ ( P2 )   uu r uu r   n2 a2 qua A ( ...) ( P2 ) :   uu r ⇔ ( P2 ) : ... vtptn2   ptmp ( P )  Kết luận: Phương trình giao tuyến (d) của (P1) và (P2) có dạng: ( d ) :  1  ptmp ( P2 )  2. BÀI TOÁN 2: “Lập phương trình đường thẳng đi qua A, cắt hai đường thẳng (d1) và (d2)” Bước 1:
  2. sivantran@gmail.com - 01689583116 Cách 1: Sử dụng pp chùm mặt phẳng : -Gọi (P) là mặt phẳng qua A chứa (d1), ta có (P) thuộc chùm tạo bởi (d1), có dạng : (P) : m(pt(1) của (d1)) + n(pt2 của (d1)) = 0 ⇔ ( P ) :...... uuuu r Cách 2: Chọn điểm M(...) tùy ý thuộc (d1) ⇒ AM ( ...) qua A ( ...) qua A ( ...)  r uuuu  r r uuuu ur r ( P) :   ⇔ ( P2 ) : n AM ⇔ n =  AM .a1  = ....   ( d 2 ) ∈ ( P )   r ur   n a1 qua A ( ...) ( P) :  r  ⇔ ( P ) : ... vtptn  Bước 2: Gọi B là giao điểm của (P) và (d2). Khi đó tọa độ của B là nghiệm của hệ:  pt1 of ( d 2 ) x =    pt2 of ( d 2 ) ⇒  y = ⇒ B (...)  z =  pt ( P )  Chú ý: nếu không tồn tại B. Kết luận bài toán vô nghiệm Nếu có vô số nghiệm. Kết luận bài toán có vô số nghiệm đó chính là chùm đường thẳng chứa (d) đi qua A. Bước 3: Gọi (d) là đường thẳng qua A, B, ta có:  x = ... qua A ( ....)   ( d ) :  uuu r ⇔ ( d ) :  y = ..., t ∈ R. vtcp AB ( ....)   z = ...  ur ur Gọi a1 là vtcp của (d1), ta có a1 ( ...) ur uuu r Từ đó, dễ thấy a1 không cùng phương với AB. Vậy, (d): ... là đường thẳng cần dựng. 3. BÀI TOÁN 3: “Lập phương trình đường thẳng (d1) qua A, vuông góc với (d) và nằm trong mặt phẳng Bước 1: - Kiểm tra (d) có cắt (P) tại A không. - Lập phương trình mặt phẳng (Q) thỏa mãn: qua A ( ....) qua A ( ....) ( Q) :    ⇔ ( Q)  uu r  ( Q) ( d )   vtptad ( ....) Bước 2: Khi đó đường thẳng (d1) chính là giao tuyến của (P) và (Q). 4. BÀI TOÁN 4: “Lập phương trình đường thẳng (d1) qua A, vuông góc với (d) và cắt (d)”
  3. sivantran@gmail.com - 01689583116 Gọi (d1) là đường thẳng qua A vuông góc với (d) và cắt (d), vậy (d1) qua A và H (H là hình chiếu vuông góc của A lên (d). * Xác định H: r r Gọi a là vtcp của (d), ta có a ( ...)  x = ... Chuyển phương trình (d) về dạng tham số: ( d ) :  y = ..., t ∈ R.   z = ...  uuur Vì H ∈ ( d ) , nên H (theo t) ⇒ AH ( ...) uuur r AH ( d ) ⇔ AH .a = 0 ⇔ ... ⇔ t = ... ⇒ H ( ...) Phương trình (d1), được xác định bởi: qua A ( ....) ( d1 ) :   uuur ⇔ ( d1 ) :... vtcp AH ( ....)  Dựng (P1) và (P2) thỏa mãn: qua A ( ...) qua A ( ...) ( P1 ) :    và ( P2 ) :  ( P )  1 ( d) ( d ) ∈ ( P2 )  Khi đó ( d1 ) = ( P1 ) ∩ ( P2 ) 5. BÀI TOÁN 5: “Xác định tọa độ hình chiếu vuông góc của điểm A lên mặt phẳng (P) r Mặt phẳng (P) có vtpt n ( ...) Gọi (d) là đường thẳng qua A và vuông góc với (P), ta được:  qua A ( ....)   ( d) : r ⇔ ( d ) :  ,t ∈ R   vtcpn ( ....)   Vì hình chiếu vuông góc H của A lên (P) chính là giao điểm của (d) và (P), do đó: thay các tọa độ của (d) vào (P) ⇔ t = ... ⇒ H ( ...) 6. BÀI TOÁN 6: “Xác định tọa độ điểm A1 đối xứng với A qua mặt phẳng (P) Bước 1: Xác định tọa độ hình chiếu vuông góc H của A lên mặt phẳng (P). Bước 2: Suy ra tọa độ điểm A1 từ điều kiện H là trung điểm của AA1. 7. BÀI TOÁN 7: “Xác định tọa độ hình chiếu vuông góc của điểm A lên đường thẳng (d)
  4. sivantran@gmail.com - 01689583116 Cách 1: r r Gọi a là vtcp của (d), ta có a ( ...)  x = ... Chuyển phương trình (d) về dạng tham số: ( d ) :  y = ..., t ∈ R.   z = ...  uuur Vì H ∈ ( d ) , nên H (theo t) ⇒ AH ( ...) uuur r AH ( d ) ⇔ AH .a = 0 ⇔ ... ⇔ t = ... ⇒ H ( ...) Cách 2: r r Gọi a là vtcp của (d), ta có a ( ...) Gọi H(x,y,z) là hình chiếu vuông góc của A lên đường thẳng (d), suy ra: H ∈ ( d )  H ∈ ( d )  H ∈ ( d )   ⇔  uuu r ⇔  uuu r r r ⇒   AH ( d ) AH a   AH.a = 0  8. BÀI TOÁN 8: “Xác định tọa độ điểm A1 đối xứng với A qua đường thẳng (d) Bước 1: Xác định tọa độ hình chiếu vuông góc H của A lên đường thẳng (d). Bước 2: Suy ra tọa độ điểm A1 từ điều kiện H là trung điểm của AA1. x +1 y −1 z − 3 x y −1 z − 3 Bài 1: Cho (d1) là đường thẳng: = = và đường thẳng (d2): = = . 3 2 −2 1 1 2 Lập phương trình mặt phẳng chứa (d1) và (d2). ĐS: 6x-8y+z+11=0 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 2: Lập phương trình đường thẳng đi qua điểm M(-1, 2, -3), vuông góc với vectơ r a = (6; 2; −3) và cắt đường thẳng:  x = 1 + 3t ( d ) :  y = −1 + 2t , t ∈ R.   z = 3 − 5t  x −1 y +1 z − 3 ĐS: = = 3 2 −5
  5. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 3: Tìm phương trình đường thẳng đi qua điểm (1; 2; -2) và song song với đường thẳng: x + y − z + 2 = 0  2 x − y + 5 z − 1 = 0 x −1 y − 2 z + 2 ĐS: = = 4 −7 −3 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... r Bài 4: Trong không gian Oxyz cho điểm A(-1, 2, 3); a = (6; −2; −3) và đường thẳng (d) có 2 x − 3 y − 5 = 0 phương trình  5 x + 2 z − 14 = 0 a) Lập phương trình mặt phẳng ( α ) chứa A và (d). r B)Lập phương trình đường thẳng ( ∆ ) đi qua A và vuông góc với vectơ a và cắt đường thẳng (d). x +1 y − 2 z − 3 ĐS: ( α ) : 3x+3y+2z-9=0; ( ∆ ) : = = 5 −21 24 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
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Bài 5: Trong không gian Oxyz cho điểm A(2, -1, 1); và đường thẳng y + z − 4 = 0 ( ∆) :  2 x − y − z + 2 = 0 a) Viết phương trình mặt phẳng ( α ) đi qua A và vuông góc với ( ∆ ) . b) Xác định tọa độ điểm B đối xứng với A qua ( ∆ ) . ĐS: ( α ) : y − z + 2 = 0 ; B(0; 3; 5) ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 6: Trong không gian Oxyz cho điểm A(3, 2, 1); và đường thẳng: x y ( d) : = = z +3 2 4
  7. sivantran@gmail.com - 01689583116 a)Viết phương trình mặt phẳng ( P ) đi qua A và chứa (d) . b) Viết phương trình đường thẳng ( ∆ ) đi qua A, vuông góc với (d) và cắt (d). 14 x − 5 y − 8 z − 24 = 0 ĐS: (P): 14x-5y-8z-24=0; ( ∆ ) :  2 x + 4 y + z − 15 = 0 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 7: Viết phương trình đường thẳng ( ∆ ) đi qua M(1; 1; 2) và song song với đường thẳng: 3x − y + 2 z − 7 = 0 ( d) : x + 3 y − 2z + 3 = 0 x −1 y −1 z − 2 ĐS: ( ∆ ) : = = −2 4 5 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 8: Trong không gian Oxyz cho đường thẳng (d) và mặt phẳng (P) có phương trình:  x = 1 + 2t ( d ) :  y = 2 − t , t ∈ R.  ( P) : 2x − y − 2z +1 = 0  z = 3t  a) Tìm tọa độ các điểm thuộc đường thẳng (d) sao cho khoảng cách từ mỗi điểm đó đến mặt phẳng (P) bằng 1.
  8. sivantran@gmail.com - 01689583116 b) Gọi K là điểm đối xứng của điểm I(2; -1; 3) qua đường thẳng (d). Hãy xác định tọa độ điểm K. ĐS: M1(-3; 4; -6) và M2(9; -2; 12); K(4; 3; 3). ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 9: Trong không gian Oxyz cho ba điểm A(1; 3; 2), B(1; 2; 1), C(1; 1; 3). Hãy viết phương trình tham số của đường thẳng (d) đi qua trọng tâm tam giác ABC và vuông góc với mặt phẳng chứa tam giác đó. x = 1+ t  ĐS: ( d ) :  y = 2 , t ∈ R. z = 2  ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 10: Viết phương trình đường thẳng đi qua điểm M(2; -1; 0), vuông góc và cắt đường thẳng (d) có phương trình:
  9. sivantran@gmail.com - 01689583116 5 x + y + z + 2 = 0 ( d) : x − y + 2z +1 = 0 x − 2 y +1 z ĐS: ( ∆ ) : = = 2 0 1 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 11: Trong không gian Oxyz cho mặt phẳng (P): 3x+6y-z-2=0, và đường thẳng:  x + y − 7 z − 14 = 0 ( d) : x − y − z − 2 = 0 a) Tìm tọa độ giao điểm A của (P) và (d). b) Tìm phương trình mặt phẳng ( β ) qua B(1; 2; -1) và vuông góc với (d). ĐS: A(0; 0; -2); ( β ) : 4x+3y+z-9=0 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 12: Trong không gian Oxyz cho đường thẳng (d) và mặt phẳng (P) có phương trình:
  10. sivantran@gmail.com - 01689583116 x = 1+ t ( d ) :  y = t − 1, t ∈ R.  ( P ) : x + 2 y + z −1 = 0  z = 2t  a) Tìm tọa độ các điểm thuộc đường thẳng (d) sao cho khoảng cách từ mỗi điểm đó đến mặt phẳng (P) bằng 6 . b) Tìm tọa độ của điểm N đối xứng với điểm M(2; 0; -1) qua đường thẳng (d).  13 3 16  1 9 8 ĐS: A1  ; ;  ; A2  ; − ; −  ; N ( 0; −2;1) 5 5 5 5 5 5 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... x − 2z = 0 Bài 13: Lập phương trình mặt phẳng chứa đường thẳng:  và vuông góc 3 x − 2 y + z − 3 = 0 với mặt phẳng: x – 2y + z + 5 = 0. ĐS: ( β ) : 11x – 2y -15z – 3 = 0 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
  11. sivantran@gmail.com - 01689583116 x +1 y −1 z − 3 x y −1 z − 3 Bài 14: Cho đường thẳng ( d1 ) : = = và đường thẳng ( d 2 ) : = = . 3 2 −2 1 1 2 Tìm tọa độ giao điểm của (d1) và (d2). ĐS: A(2; 3; 1) ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 15: Chứng minh rằng hai đường thẳng sau đây cắt nhau:  x = 2t  x = t '+ 5 ( d1 ) :  y = 3t − 2, t ∈ R; ( d 2 ) :  y = −4t '− 1, t ' ∈ R    z = 4t + 6  z = t '+ 20   ĐS: M(3; 7; 18) ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 16: Chứng tỏ rằng hai đường thẳng sau đây không cắt nhau và vuông góc nhau: x y −1 z 3x + y − 5 z + 1 = 0 ( d1 ) : = = ; ( d2 ) :  1 −2 3 2 x + 3 y − 8 z + 1 = 0 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
  12. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 17: Viết phương trình chính tắc của đường thẳng (d) qua M(1; 5; 0) và cắt cả hai 2 x − z − 1 = 0 3 x + y − 2 = 0 đường thẳng: ( d1 ) :  ; ( d2 ) :  x + y − 4 = 0 y − z − 2 = 0 x −1 y − 5 z ĐS: = = 1 3 0 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 18: Viết phương trình chính tắc của đường thẳng (d) qua A(0; 1; 1), vuông góc với x −1 y + 2 z thẳng: ( d1 ) : = = và cắt đường thẳng Viết phương trình chính tắc của đường 3 1 1 x + y − z + 2 = 0 thẳng (d) qua M(1; 5; 0) và cắt cả hai đường thẳng: ( d 2 ) :  x +1 = 0 x −1 y −1 z −1 ĐS: = = 1 −1 −2 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 19: Lập phương trình đường thẳng đi qua điểm A(-1, 2, -3), vuông góc với vectơ r a = (6; −2; −3) và cắt đường thẳng (d):
  13. sivantran@gmail.com - 01689583116 x −1 y +1 z − 3 ( d) : = = 3 2 −5 x −1 y +1 z − 3 ĐS: = = 2 −3 6 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 20: Cho 2 đường thẳng 2 x − 3 y − 2 = 0 2 x − 3 y + 9 = 0 ( d1 ) :  ; ( d2 ) :   x + 3z + 2 = 0  y + 2z +1 = 0 a) Chứng minh (d1)//(d2). Viết phương mặt phẳng chứa (d1) và (d2). b) Tìm tọa độ điểm N đối xứng với M(-2; 3; -4) qua (d1) ĐS: x + 4y + 11z +10 = 0; N(4; -3; 2). ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
  14. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 21: Cho điểm A(0; 1; 1) và 2 đường thẳng x −1 y + 2 z x + y − z + 2 = 0 ( d1 ) : = = ; ( d2 ) :  3 1 1 x +1 = 0 Lập phương trình đường thẳng qua A, vuông góc (d1) và cắt (d2). x y −1 z −1 ĐS: = = −1 1 2 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 22: Trong không gian cho 2 đường thẳng x −7 y −3 z −9 x − 3 y −1 z −1 ( d1 ) : = = ; ( d2 ) : = = 1 2 −1 −7 2 3 Chứng tỏ rằng hai đường thẳng đó chéo nhau. ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 23: Cho đường thẳng xác định bởi phương trình: x+2 y+2 z = = và điểm M(4; -3; 2). Tìm tọa độ điểm N là hình chiếu vuông góc của 3 2 −1 điểm M lên đường thẳng đã cho. ĐS: N(1; 0; -1) ...................................................................................................................................................... ......................................................................................................................................................
  15. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... x − 2 y −1 z Bài 24: Cho điểm A(1; 0; 0) và đường thẳng (d): ( d ) : = = 1 2 1 a) Viết phương trình mặt phẳng đi qua A và vuông góc với (d) b) Tính khoảng cách từ A đến (d). 2 ĐS: x + 2y + z – 1 = 0; 2 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... x y −1 Bài 25: Cho điểm A(1; 2; 1) và đường thẳng (d): ( d ) : = = z+3 3 4 a) Viết phương trình mặt phẳng đi qua A và chứa đường thẳng (d) b) Tính khoảng cách từ A đến đường thẳng (d). 347 ĐS: 15x – 11y –z + 8 = 0; 26 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
  16. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 26: Trong không gian cho 2 đường thẳng  x = 2t + 1 x = 2 + s ( d1 ) :  y = t + 2 và ( d 2 ) :  y = −3 + 2s    z = 3t − 2  z = 1 + 3s   a) Chứng tỏ rằng (d1) và (d2) chéo nhau. b) Tính khoảng cách giữa (d1) và (d2). 8 3 ĐS: 3 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 27: Trong không gian cho 2 đường thẳng x −7 y −3 z −9 x − 3 y −1 z −1 ( d1 ) : = = và ( d 2 ) : = = 1 2 −1 −7 2 3 a) Chứng tỏ rằng (d1) và (d2) chéo nhau. b) Tính khoảng cách giữa (d1) và (d2). ĐS: 2 21 ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
  17. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 28: Trong không gian cho 2 đường thẳng  x = −2 + 2t  x + y + 2z = 0  ( d1 ) :  và ( d 2 ) :  y = −5t ,t ∈ R x − y + z +1 = 0 z = 2 + t  a) Chứng tỏ rằng (d1) và (d2) chéo nhau. b) Tính khoảng cách giữa (d1) và (d2). 17 ĐS: 419 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... Bài 29: Trong không gian cho 2 đường thẳng x − 2 y z +1 x +1 y − 2 z ( d1 ) : = = và ( d 2 ) : = = 1 −1 −2 2 1 −1 a) Tính khoảng cách giữa (d1) và (d2). b) Tìm tọa độ điểm A đối xứng với điểm B(3; -3; 2) qua đường thẳng (d1). 4  1 11 8  ĐS: ; A ; ; −  3 3 3 3 ...................................................................................................................................................... ......................................................................................................................................................
  18. sivantran@gmail.com - 01689583116 ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................

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