Hướng dẫn giải Bài 12 (Trang 93 SGK Toán Hình học 12)

12. Tìm tọa độ điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>'</mo></math> đối xứng với điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>5</mn></mrow></mfenced></math> qua đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> có phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>2</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math> Giải: Đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> có vecto chỉ phương <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>2</mn></mrow></mfenced></math> Gọi <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> là mặt phẳng qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> và vuông góc với <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> thì có vecto pháp tuyến <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>2</mn></mrow></mfenced></math> do đó phương trình mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mi>z</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇔</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>1</mn></mfenced></math> <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/19022022/34849d9e-ff38-4833-a8a8-9f61636d4253.PNG" /> Hình chiếu <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> của <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> lên <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> là giao điểm của <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math>. Thay <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>t</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><mi>t</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mn>2</mn><mi>t</mi></math> vào <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced></math> ta được: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>t</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>+</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>t</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> Khi đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mo> </mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced><mo>.</mo></math> Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mo>'</mo></msup></math> là điểm đối xứng của <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> nên: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mn>2</mn><mover><mrow><mi>A</mi><mi>H</mi></mrow><mo>→</mo></mover><mo>⇔</mo><mfenced open="{" close=""><mtable><mtr><mtd><msub><mi>x</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo>.</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>-</mo><mn>1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mi>y</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>+</mo><mn>2</mn><mo>=</mo><mn>2</mn><mo>.</mo><mfenced><mrow><mn>0</mn><mo>+</mo><mn>2</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mi>z</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>+</mo><mn>5</mn><mo>=</mo><mn>2</mn><mo>.</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mn>5</mn></mrow></mfenced></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable><mtr><mtd><msub><mi>x</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><msub><mi>x</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><msub><mi>x</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> Vậy điểm đối xứng với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> qua đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> là <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>'</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced><mo>.</mo></math>