Hướng dẫn giải Bài 10 (Trang 93 SGK Toán Hình học 12)

10. Cho điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mfenced><mrow><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> và mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced><mo>:</mo><mo> </mo><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>-</mo><mn>27</mn><mo>=</mo><mn>0</mn></math>. Tìm tọa độ điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>'</mo></math> đối xứng với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> Giải: Gọi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> là hình chiếu của <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> lên mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math>. Gọi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> là đường thẳng đi qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> và vuông góc với <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> có vecto chỉ phương là <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>a</mi><mi>d</mi></msub><mo>→</mo></mover><mo>=</mo><mover><msub><mi>n</mi><mi>α</mi></msub><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mn>3</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> Phương trình tham số của <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>2</mn><mo>+</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mo>-</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math> Thay <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn><mo>+</mo><mi>t</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi>t</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mo>-</mo><mi>t</mi></math> vào phương trình mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math>, ta được: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>t</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mi>t</mi></mrow></mfenced><mo>-</mo><mn>27</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mn>11</mn><mi>t</mi><mo>-</mo><mn>22</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>t</mi><mo>=</mo><mn>2</mn></math> Khi đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn><mo>;</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>7</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><mn>4</mn><mo>;</mo><mo> </mo><mn>7</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>'</mo></math> đối xứng với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> nên:  <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>M</mi><mi>M</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mn>2</mn><mover><mrow><mi>M</mi><mi>H</mi></mrow><mo>→</mo></mover><mo>⇔</mo><mfenced open="{" close=""><mtable><mtr><mtd><msub><mi>x</mi><mrow><mi>M</mi><mo>'</mo></mrow></msub><mo>-</mo><mn>2</mn><mo>=</mo><mn>2</mn><mfenced><mrow><mn>4</mn><mo>-</mo><mn>2</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mi>y</mi><mrow><mi>M</mi><mo>'</mo></mrow></msub><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn><mfenced><mrow><mn>7</mn><mo>-</mo><mn>1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mi>z</mi><mrow><mi>M</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable><mtr><mtd><msub><mi>x</mi><mrow><mi>M</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>6</mn></mtd></mtr><mtr><mtd><msub><mi>y</mi><mrow><mi>M</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>13</mn></mtd></mtr><mtr><mtd><msub><mi>z</mi><mrow><mi>M</mi><mo>'</mo></mrow></msub><mo>=</mo><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math> Vậy điểm đối xứng của điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> qua mặt phẳng <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"><mi>M</mi><mo>'</mo><mfenced><mrow><mn>6</mn><mo>;</mo><mo> </mo><mn>13</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math>.