Hướng dẫn giải Bài 14 (Trang 101 SGK Toán Hình học 12)

Trong không gian cho ba điểm A,B,C a) Xác định điểm G sao cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover><mo> </mo><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover></math><span id="MathJax-Element-1-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"><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>&#x2192;</mo></mover><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mn>0</mn><mo stretchy="false">&#x2192;</mo></mover></mrow></math>"> b) Tìm tập hợp các điểm M sao cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>M</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>M</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup></math>, với k là hằng số Giải a) Ta có : <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover><mo> </mo><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover><mo> </mo><mo>⇔</mo><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mn>2</mn><mo>(</mo><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>)</mo><mo> </mo><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mn>2</mn><mover><mrow><mi>C</mi><mi>B</mi></mrow><mo>→</mo></mover><mo> </mo><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover><mo> </mo><mo>⇔</mo><mover><mrow><mi>A</mi><mi>G</mi></mrow><mo>→</mo></mover><mo>=</mo><mn>2</mn><mover><mrow><mi>C</mi><mi>B</mi></mrow><mo>→</mo></mover></math> Suy ra G nằm trên đường thẳng qua A và song song với CB sao cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>G</mi></mrow><mo>→</mo></mover><mo>=</mo><mn>2</mn><mover><mrow><mi>C</mi><mi>B</mi></mrow><mo>→</mo></mover></math> b) Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mover><mrow><mi>M</mi><mi>A</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mover><mrow><mi>M</mi><mi>B</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mover><mrow><mi>M</mi><mi>C</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><msup><mfenced><mrow><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>M</mi><mi>G</mi></mrow><mo>→</mo></mover></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mfenced><mrow><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>G</mi><mi>M</mi></mrow><mo>→</mo></mover></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mfenced><mrow><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>G</mi><mi>M</mi></mrow><mo>→</mo></mover></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><msup><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>+</mo><msup><mover><mrow><mi>G</mi><mi>M</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo> </mo><mo>-</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>M</mi></mrow><mo>→</mo></mover><mfenced><mrow><mover><mrow><mi>G</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mn>2</mn><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><msup><mover><mrow><mi>M</mi><mi>G</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>G</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>G</mi><msup><mi>C</mi><mn>2</mn></msup></mrow></mfenced></math> Từ đó suy ra: <ul> <li>Nếu <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>G</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>G</mi><msup><mi>C</mi><mn>2</mn></msup></mrow></mfenced><mo><</mo><mn>0</mn></math> thì M không tồn tại.</li> <li>Nếu <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>G</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>G</mi><msup><mi>C</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mi>t</mi><mi>h</mi><mi>ì</mi><mo> </mo><mi>M</mi><mo>≡</mo><mi>G</mi></math></li> <li>Nếu <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>G</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>G</mi><msup><mi>C</mi><mn>2</mn></msup></mrow></mfenced><mo>></mo><mn>0</mn></math> thì tập hợp các điểm M là đường tròn tâm tâm G, bán kính bằng <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>G</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>G</mi><msup><mi>C</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math></li> </ul>

Hướng dẫn Giải Bài 14 (trang 100, SGK Toán 12, Hình học)

GV:

GV colearn

Xem lời giải bài tập khác cùng bài

Hướng dẫn giải Bài 1 (Trang 99 SGK Toán Hình học 12)