Hướng dẫn giải Bài 8 (Trang 93 SGK Toán Hình học 12)

8. Viết phương trình mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> tiếp xúc với mặt cầu <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced><mo>:</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>26</mn><mi>z</mi><mo>+</mo><mn>170</mn><mo>=</mo><mn>0</mn></math> và song song với hai đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>:</mo><mo> </mo><mfenced open="{" close=""><mrow><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>5</mn><mo>+</mo><mn>2</mn><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><mn>13</mn><mo>+</mo><mn>2</mn><mi>t</mi></mtd></mtr></mtable><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>;</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>d</mi><mo>'</mo><mo>:</mo><mo> </mo><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>7</mn><mo>+</mo><mn>3</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> Giải: Đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>'</mo></math> lần lượt có vecto chỉ là <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>3</mn><mo>;</mo><mo> </mo><mn>2</mn></mrow></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>a</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>3</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. Mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> song song với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>'</mo></math> có vecto pháp tuyến <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mover><mi>a</mi><mo>→</mo></mover><mo>,</mo></mtd><mtd><mover><mrow><mi>a</mi><mo>'</mo></mrow><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>4</mn><mo>;</mo><mo> </mo><mn>6</mn><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> có dạng: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>+</mo><mi>D</mi><mo>=</mo><mn>0</mn></math> Mặt cầu <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced></math> có tâm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mfenced><mrow><mn>5</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>13</mn></mrow></mfenced></math> và bán kính <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><mi>d</mi></msqrt><mo>=</mo><msqrt><mn>25</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>169</mn><mo>-</mo><mn>170</mn></msqrt><mo>=</mo><mn>5</mn></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> tiếp xúc với <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced></math>  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mi>d</mi><mfenced><mrow><mi>I</mi><mo>,</mo><mo> </mo><mfenced><mi>α</mi></mfenced></mrow></mfenced><mo>=</mo><mi>R</mi><mspace linebreak="newline"/><mo>⇔</mo><mfrac><mfenced open="|" close="|"><mrow><mn>4</mn><mo>.</mo><mfenced><mn>5</mn></mfenced><mo>+</mo><mn>6</mn><mo>.</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>5</mn><mo>.</mo><mfenced><mrow><mo>-</mo><mn>13</mn></mrow></mfenced><mo>+</mo><mi>D</mi></mrow></mfenced><msqrt><mn>16</mn><mo>+</mo><mn>36</mn><mo>+</mo><mn>25</mn></msqrt></mfrac><mo>=</mo><mn>5</mn><mspace linebreak="newline"/><mo>⇔</mo><mfenced open="|" close="|"><mrow><mi>D</mi><mo>-</mo><mn>51</mn></mrow></mfenced><mo>=</mo><mn>5</mn><msqrt><mn>77</mn></msqrt><mo>⇔</mo><mi>D</mi><mo>=</mo><mn>51</mn><mo>±</mo><mn>5</mn><msqrt><mn>77</mn></msqrt></math> Vậy ta có hai mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> thỏa mãn đề bài. Phương trình tổng quát của <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>4</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>+</mo><mn>51</mn><mo>±</mo><mn>5</mn><msqrt><mn>77</mn></msqrt><mo>=</mo><mn>0</mn><mo>.</mo></math>