Hướng dẫn giải Bài 5 (Trang 92 SGK Toán Hình học 12)

5. Cho mặt cầu <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced></math> có phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>100</mn></math> và mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> có phương trình <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math>. Mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> cắt mặt cầu <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced></math> theo một đường tròn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="script">C</mi></mfenced></math>. Hãy xác định tọa độ tâm và tính bán kính của đường tròn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="script">C</mi></mfenced></math>. Giải: Mặt cầu <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced></math> có tâm là <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mfenced><mrow><mn>3</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> và có bán kính <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>10</mn></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mrow><mi>I</mi><mo>,</mo><mo> </mo><mfenced><mi>α</mi></mfenced></mrow></mfenced><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><mn>2</mn><mo>.</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>2</mn><mo>-</mo><mn>1</mn><mo>+</mo><mn>9</mn></mrow></mfenced><msqrt><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mn>6</mn></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mrow><mi>I</mi><mo>,</mo><mo> </mo><mfenced><mi>α</mi></mfenced></mrow></mfenced><mo>=</mo><mn>6</mn><mo><</mo><mn>10</mn></math>, suy ra mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> cắt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>S</mi></mfenced></math> theo một đường tròn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="script">C</mi></mfenced><mo>.</mo></math> Tâm J của <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="script">C</mi></mfenced></math> chính là hình chiếu vuông góc của <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> trên mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math>. Đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> đi qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> và vuông góc với <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"><mo>∆</mo></math> có phương trình là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>3</mn><mo>+</mo><mn>2</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math>cắt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> tại <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mfenced><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mi>t</mi><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>t</mi><mo>;</mo><mo> </mo><mn>1</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>.</mo></math> Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo>∈</mo><mfenced><mi>α</mi></mfenced></math> nên ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mi>t</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>t</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>t</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo></math> Vậy ta được <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>. Bán kính <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> của <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="script">C</mi></mfenced></math> được tính theo công thức: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><msqrt><msup><mi>R</mi><mn>2</mn></msup><mo>-</mo><msup><mi>d</mi><mn>2</mn></msup><mfenced><mrow><mi>I</mi><mo>,</mo><mo> </mo><mfenced><mi>α</mi></mfenced></mrow></mfenced></msqrt><mo>=</mo><msqrt><mn>100</mn><mo>-</mo><mn>36</mn></msqrt><mo>=</mo><mn>8</mn><mo>.</mo></math> Vậy đường tròn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="script">C</mi></mfenced></math> có tâm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> và bán kính <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>8</mn></math>