Hướng dẫn giải Bài 9 (Trang 100 SGK Toán Hình học 12)

Trong không gian Oxyz cho bốn điểm<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>A</mi><mo>(</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo> </mo><mi>B</mi><mo>(</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo> </mo><mi>C</mi><mo>(</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>;</mo><mo> </mo><mn>3</mn><mo>)</mo><mo>,</mo><mo> </mo><mi>D</mi><mo>(</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>.</mo></math> a) Chứng mỉnh rằng các đường thẳng AB, AC, AD vuông góc với nhau từng đôi một. Tính thể tích tứ diện ABCD. b) Viết phương tình mặt cầu (S) đi qua bốn điểm A, B, C, D. c) Viết phương trình mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo></math> tiếp xúc với mặt cầu (S) và song song mặt phẳng (ABD). Giải: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/98faf747-672a-4144-9035-a6bb16b43c47.PNG" /> a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>)</mo><mo>;</mo><mo> </mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>)</mo><mo>;</mo><mo> </mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>0</mn><mo>;</mo><mo>-</mo><mn>2</mn><mo>;</mo><mn>0</mn><mo>)</mo></math> Ta có : <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>=</mo><mn>0</mn></math> Suy ra AB, AC, AD vuông góc với nhau từng đôi một. Thể tích tứ diện ABCD là: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>S</mi><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></msub><mo>.</mo><mi>A</mi><mi>D</mi><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>.</mo><mi>A</mi><mi>B</mi><mo>.</mo><mi>A</mi><mi>C</mi><mo>.</mo><mi>A</mi><mi>D</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>.</mo></math> b) Gọi M là trung điểm BC ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>(</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>;</mo><mn>4</mn><mo>;</mo><mn>1</mn><mo>)</mo></math> Gọi d là đường thẳng đi qua M và vuông góc với mặt phẳng (ABC) ta suy ra d chính là trục đường tròn ngoại tiếp tam giác ABC. Gọi (P) là mặt phẳng trung trực của đường thẳng AD. Khi đó giao điểm của (P) và d chính là tâm đường tròn ngoại tiếp tứ diện ABCD. Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>M</mi><mi>I</mi></mrow><mo>→</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced open="{" close=""><mtable><mtr><mtd><msub><mi>x</mi><mi>I</mi></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mn>0</mn><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><msub><mi>y</mi><mi>I</mi></msub><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mo>(</mo><mo>-</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd><msub><mi>z</mi><mi>I</mi></msub><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mn>0</mn><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> Vậy tâm mặt cầu (S) là <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo> </mo><mo>(</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>;</mo><mn>3</mn><mo>;</mo><mn>1</mn><mo>)</mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>,</mo></math>bán kính  Vậy phương trình mặt cầu (S) ngoại tiếp tứ diện ABCD là: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>3</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><mfrac><mn>21</mn><mn>4</mn></mfrac></math> c) Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>)</mo><mo>;</mo><mo> </mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>0</mn><mo>;</mo><mo>-</mo><mn>2</mn><mo>;</mo><mn>0</mn><mo>)</mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi></mi></mrow></mover></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo></mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>,</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mo>(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>2</mn><mo>)</mo></math> Mặt phẳng (ABD) có một VTPT là:<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mfenced open="[" close="]"><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>,</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo> </mo><mo>=</mo><mfenced><mrow><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>1</mn></mrow></mfenced></math> Do <span id="MathJax-Element-23-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> song song với (ABD) nên cũng nhận <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>1</mn></mrow></mfenced></math> làm VTPT Suy ra phương trình mặt phẳng <span id="MathJax-Element-25-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> có dạng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo><mo> </mo><mi>z</mi><mo>+</mo><mi>D</mi><mo>=</mo><mn>0</mn></math>. <span id="MathJax-Element-27-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> tiếp xúc với mặt cầu <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>d</mi><mo>(</mo><mi>I</mi><mo>,</mo><mo>(</mo><mo> </mo><mi>α</mi><mo> </mo><mo>)</mo><mo>)</mo><mo>=</mo><mi>r</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mo>|</mo><mn>1</mn><mo>+</mo><mi>D</mi><mo>|</mo><mo>=</mo><mfrac><msqrt><mn>21</mn></msqrt><mn>2</mn></mfrac><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>D</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>21</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>D</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><mfrac><msqrt><mn>21</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mo> </mo></math> Vậy có hai mặt phẳng <span id="MathJax-Element-30-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>">(α) thỏa mãn đề bài:  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><msub><mi>α</mi><mn>1</mn></msub><mo>)</mo><mo>:</mo><mo> </mo><mi>z</mi><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>21</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><msub><mi>α</mi><mn>2</mn></msub><mo>)</mo><mo>:</mo><mo> </mo><mi>z</mi><mo>−</mo><mn>1</mn><mo>-</mo><mfrac><msqrt><mn>21</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math>

Hướng dẫn Giải Bài 9 (trang 100, SGK Toán 12, Hình học)

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