Hướng dẫn giải Câu 6 (Trang 95 SGK Toán Hình học 12)

Mặt cầu ngoại tiếp tứ diện ABCD có bán kính là: (A) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math>;         (B) <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt></math>;         (C) <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt></math>;         (D) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>4</mn></mfrac></math>. Giải <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/21022022/2-PxhWTO.png" /> Mặt cầu đi qua 4 điểm A, B, C, D có dạng (S): <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>-</mo><mn>2</mn><mi>b</mi><mi>y</mi><mo>-</mo><mn>2</mn><mi>c</mi><mi>z</mi><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></math>. vì A, B, C, D <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈</mo></math> (S) nên:  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mn>2</mn><mi>c</mi><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>-</mo><mn>2</mn><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced></math> Bán kính mặt cầu (S) là <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></math> =<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math>. Chọn (A)