Hướng dẫn giải Bài 10 (Trang 100 SGK Toán Hình học 12)

Trong không gian Oxyz cho đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>d</mi></mfenced><mo>:</mo><mo> </mo><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mn>2</mn><mo>+</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math>và mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo><mo>:</mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> a) Tìm toạ độ giao điểm A của (d) và <span id="MathJax-Element-3-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). b) Viết phương trình mặt phẳng <span id="MathJax-Element-4-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>&#x03B2;</mi><mo stretchy="false">)</mo></math>">(β) qua A và vuông góc với (d). Giải: a) Toạ độ giao điểm A của d và <span id="MathJax-Element-7-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-table; 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) là nghiệm của hệ phương trình. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi>t</mi></mtd></mtr></mtable><mo>(</mo><mn>1</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mn>2</mn><mo>+</mo><mi>t</mi><mo> </mo><mo>(</mo><mn>2</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>t</mi><mo> </mo><mo>(</mo><mn>3</mn><mo>)</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo> </mo><mo>(</mo><mn>4</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math> Thay (1), (2), (3) và (4) ta được: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>t</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>+</mo><mi>t</mi><mo>+</mo><mn>3</mn><mo>−</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo>⇔</mo><mo>−</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>t</mi><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac></math> Khi đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>10</mn><mn>4</mn></mfrac><mo>;</mo><mi>y</mi><mo>=</mo><mfrac><mn>15</mn><mn>4</mn></mfrac><mo>;</mo><mi>z</mi><mo>=</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo><mfenced><mrow><mo>-</mo><mfrac><mn>10</mn><mn>4</mn></mfrac><mo>;</mo><mfrac><mn>15</mn><mn>4</mn></mfrac><mo>;</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></mfenced></math> b) Đường thẳng d có vecto chỉ phương là <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>a</mi><mi>d</mi></msub><mo>→</mo></mover><mo>=</mo><mo> </mo><mo>(</mo><mo>−</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo>−</mo><mn>1</mn><mo>)</mo></math> Mp <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo> </mo><mi>β</mi><mo> </mo><mo>)</mo></math> qua A vuông góc với đường thẳng d thì <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo> </mo><mi>β</mi><mo> </mo><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><mover><msub><mi>a</mi><mi>d</mi></msub><mo>→</mo></mover><mo>=</mo><mo> </mo><mo>(</mo><mo>−</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo>−</mo><mn>1</mn><mo>)</mo></math> Vậy mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo> </mo><mi>β</mi><mo> </mo><mo>)</mo></math> có phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>2</mn><mo>(</mo><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>10</mn><mn>4</mn></mfrac><mo>)</mo><mo>+</mo><mn>1</mn><mo> </mo><mo>(</mo><mo> </mo><mi>y</mi><mo>−</mo><mfrac><mn>15</mn><mn>4</mn></mfrac><mo>)</mo><mo>−</mo><mn>1</mn><mo>(</mo><mi>z</mi><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>)</mo><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mi>z</mi><mo>−</mo><mfrac><mn>30</mn><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>⇔</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>15</mn><mo>=</mo><mn>0</mn></math>

Hướng dẫn Giải Bài 10 (trang 100, SGK Toán 12, Hình học)

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