Hướng dẫn giải Bài 16 (Trang 102 SGK Toán Hình học 12)

Trong không gian Oxyz cho mặt phẳng (α) có phương trình 4x+y+2z+1 =0 và mặt phẳng (β) có phương trình 2x – 2y + z + 3 = 0 a) Chứng minh rằng <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: 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>">(α) cắt <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: 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>&#x03B2;</mi><mo stretchy="false">)</mo></math>">(β). b) Viết phương trình tham số của đường thẳng d là giao của (α) và <span id="MathJax-Element-5-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>&#x03B2;</mi><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo></math>. c) Tìm điểm M' là ảnh của M(4; 2; 1) qua phép đối xứng qua mặt phẳng <span id="MathJax-Element-6-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>">(α). d) Tìm điểm N' là ảnh của N(0; 2; 4) quá phép đối xứng qua đường thẳng d. Giải  a) Mp (α) có vectơ pháp tuyến <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mi>α</mi></msub><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>4</mn><mo>;</mo><mn>1</mn><mo>;</mo><mn>2</mn><mo>)</mo></math> Mp (β) có vectơ pháp tuyến <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mi>β</mi></msub><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>2</mn><mo>;</mo><mo>-</mo><mn>2</mn><mo>;</mo><mn>1</mn><mo>)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mi>α</mi></msub><mo>→</mo></mover><mo>,</mo><mover><msub><mi>n</mi><mi>β</mi></msub><mo>→</mo></mover><mo> </mo><mo> </mo></math>không cùng phương nên (α) cắt (β). b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mover><msub><mi>n</mi><mi>α</mi></msub><mo>→</mo></mover><mo>,</mo><mover><msub><mi>n</mi><mi>β</mi></msub><mo>→</mo></mover></mrow></mfenced><mo> </mo><mo>=</mo><mfenced><mrow><mn>5</mn><mo>;</mo><mn>0</mn><mo>;</mo><mo>−</mo><mn>10</mn></mrow></mfenced><mo>=</mo><mn>5</mn><mo>(</mo><mn>1</mn><mo>;</mo><mn>0</mn><mo>;</mo><mo>−</mo><mn>2</mn><mo>)</mo></math> Gọi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo> </mo><mo>=</mo><mo> </mo><mi>α</mi><mo> </mo><mo>∩</mo><mo> </mo><mi>β</mi></math> Vectơ chỉ phương của d vuông góc với <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mi>α</mi></msub><mo>→</mo></mover><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mover><msub><mi>n</mi><mi>β</mi></msub><mo>→</mo></mover><mo> </mo><mo> </mo></math> Nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>a</mi><mi>d</mi></msub><mo>→</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mfenced open="[" close="]"><mrow><mover><msub><mi>n</mi><mi>α</mi></msub><mo>→</mo></mover><mo>,</mo><mover><msub><mi>n</mi><mi>β</mi></msub><mo>→</mo></mover><mo> </mo></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>;</mo><mn>0</mn><mo>;</mo><mo>−</mo><mn>2</mn></mrow></mfenced></math> Tìm điểm M trên d cho x = 0 ta tìm y, z từ hệ: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>ậ</mi><mi>y</mi><mo> </mo><mi>M</mi><mo>(</mo><mn>0</mn><mo>;</mo><mn>1</mn><mo>;</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>∈</mo><mi>d</mi></math> Phương trình tham số của d là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math> c) Phương trình của đường thẳng <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"><mi mathvariant="normal">&#x0394;</mi></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi></math> đi qua M và vuông góc với <span id="MathJax-Element-24-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> là <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mn>4</mn><mo>+</mo><mn>4</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>1</mn><mo>+</mo><mn>2</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math> Để tìm giao điểm của <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>o</mi></msub></math> của Δ với (α) ta giải phương trình <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>(</mo><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>)</mo><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mo>+</mo><mo> </mo><mi>t</mi><mo> </mo><mo>+</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>t</mi><mo>)</mo><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mn>21</mn><mi>t</mi><mo> </mo><mo>+</mo><mn>21</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi>t</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>u</mi><mi>y</mi><mo> </mo><mi>r</mi><mi>a</mi><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mi>y</mi><mo>=</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mi>z</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>o</mi></msub><mfenced><mrow><mn>0</mn><mo>;</mo><mn>1</mn><mo>;</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> Vì M' là điểm đối xứng của M qua <span id="MathJax-Element-28-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> nên: MM' = 2<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>o</mi></msub></math> suy ra M'(-4; 0 ;-3) d) Mặt phẳng (γ) qua N và vuông góc với d có phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>-</mo><mn>2</mn><mo>(</mo><mi>z</mi><mo>-</mo><mn>4</mn><mo>)</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi>z</mi><mo> </mo><mo>+</mo><mo> </mo><mn>8</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math>  Để tìm giao điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>o</mi></msub></math> của d và (γ) ta giải phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mo>(</mo><mo>-</mo><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi>t</mi><mo>)</mo><mo> </mo><mo>+</mo><mo> </mo><mn>8</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mn>5</mn><mi>t</mi><mo> </mo><mo>+</mo><mo> </mo><mn>10</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi>t</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mi>h</mi><mi>i</mi><mo> </mo><mi>đ</mi><mi>ó</mi><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo>.</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>ậ</mi><mi>y</mi><mo> </mo><msub><mi>N</mi><mi>o</mi></msub><mo>(</mo><mo>-</mo><mn>2</mn><mo>;</mo><mn>1</mn><mo>;</mo><mn>3</mn><mo>)</mo></math> Vì N' là điểm đối xứng của N qua d nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>N</mi><mi>N</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mn>2</mn><mover><mrow><mi>N</mi><msub><mi>N</mi><mi>o</mi></msub></mrow><mo>→</mo></mover></math> Suy ra N'(-4; 0; 2).

Hướng dẫn Giải Bài 16 (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)