65. Find the equation of the lines through the point (3, 2) which make an angle of 45^o with the line x – 2y = 3.

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alok kumar singh | Contributor-Level 10

4 months ago

65. x – 2y = 3

y = $\frac{x}{2}$ - $\frac{3}{2}$ ______ (1)

Slope of line (1) is $\frac{1}{2} \cdot$

Let the line through P (3, 2) have slope m

Then, angle between the line = $| \frac{m - \frac{1}{2}}{1 + m \frac{1}{2}} |$

$\Rightarrow t a n 4 5^{°} = | \frac{2 m - 1}{2 + m} |$

$\Rightarrow 1 = | \frac{2 m - 1}{2 + m} |$

$\Rightarrow \frac{2 m - 1}{2 + m} = \pm 1 .$

When, $\frac{2 m - 1}{2 + m} = 1$ =>2m – 1 = 2 + m=> m = 3.

The eqn of line through (3, 2) is

y – 2 = 3 (x – 3) 3x – y – 7 = 0.

When $\frac{2 m - 1}{2 + m}$ = – 1=> 2m – 1 = – 2 – m =>3m = – 1 m = $- \frac{1}{3}$

The equation of line through (3,2) is,

y – 2 = $- \frac{1}{3}$ (x – 3) => 3y – 6 = – X + 3

x + 3y – 9 = 0

65. x – 2y = 3y = <math><mrow><mfrac><mrow><mn>x</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math> - <math><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math>______ (1)Slope of line (1) is <math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo></mrow></math>Let the line through P (3, 2) have slope mThen, angle between the line = <math><mrow><mrow><mo>|</mo><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>1</mn><mo>+</mo><mi>m</mi><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfrac></mrow><mo>|</mo></mrow></mrow></math><math><mrow><mo>⇒</mo><mi>t</mi><mi>a</mi><mi>n</mi><mn>4</mn><msup><mrow><mn>5</mn></mrow><mrow><mo>°</mo></mrow></msup><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mrow><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfrac></mrow><mo>|</mo></mrow></mrow></math><math><mrow><mo>⇒</mo><mn>1</mn><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mrow><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfrac></mrow><mo>|</mo></mrow></mrow></math><math><mrow><mo>⇒</mo><mfrac><mrow><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfrac><mo>=</mo><mo>±</mo><mn>1</mn><mo>.</mo></mrow></math>When,  <math><mrow><mfrac><mrow><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math> =>2m – 1 = 2 + m=> m = 3.The eqn of line through (3, 2) isy – 2 = 3 (x – 3) 3x – y – 7 = 0.When <math><mrow><mfrac><mrow><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfrac></mrow></math> = – 1=> 2m – 1 = – 2 – m =>3m = – 1 m = <math><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math>The equation of line through (3,2) is, y – 2 = <math><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math> (x – 3) => 3y – 6 = – X + 3x + 3y – 9 = 0

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65. Find the equation of the lines through the point (3, 2) which make an angle of 45^o with the line x – 2y = 3.

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65. Find the equation of the lines through the point (3, 2) which make an angle of 45^o with the line x – 2y = 3.