The angle between the straight lines represen | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given equation: $(x^2+y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2$ Expanding the right side: $(x^2+y^2) \sin^2 \alpha = x^2 \cos^2 \alp

Vedclass · Accepted Answer

$2\alpha$

Vedclass · Answer

(C) Given equation: $(x^2+y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2$ Expanding the right side: $(x^2+y^2) \sin^2 \alpha = x^2 \cos^2 \alpha + y^2 \sin^2 \alpha - 2xy \sin \alpha \cos \alpha$ Rearranging terms: $x^2 \sin^2 \alpha + y^2 \sin^2 \alpha = x^2 \cos^2 \alpha + y^2 \sin^2 \alpha - 2xy \sin \alpha \cos \alpha$ Simplifying: $x^2 \sin^2 \alpha = x^2 \cos^2 \alpha - 2xy \sin \alpha \cos \alpha$ $x^2(\sin^2 \alpha - \cos^2 \alpha) + 2xy \sin \alpha \cos \alpha = 0$ $-x^2 \cos(2\alpha) + xy \sin(2\alpha) = 0$ This is a homogeneous equation of the form $Ax^2 + 2Hxy + By^2 = 0$,where $A = -\cos(2\alpha)$,$H = \frac{1}{2} \sin(2\alpha)$,and $B = 0$. The angle $	heta$ between the lines is given by $	an 	heta = \left| \frac{2\sqrt{H^2 - AB}}{A+B} ight|$. Substituting values: $	an 	heta = \left| \frac{2\sqrt{\frac{1}{4} \sin^2(2\alpha) - 0}}{-\cos(2\alpha) + 0} ight| = \left| \frac{\sin(2\alpha)}{-\cos(2\alpha)} ight| = |-	an(2\alpha)| = |	an(2\alpha)|$. Thus,$	heta = 2\alpha$.

The angle between the straight lines represented by $(x^2+y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2$ is

Similar Questions

The sine of the angle between the pair of lines represented by the equation $x^2-7xy+12y^2=0$ is

The acute angle between the lines represented by $({x^2} + {y^2})\sqrt{3} = 4xy$ is

The angle between the pair of lines represented by $2x^2 - 7xy + 3y^2 = 0$ is .....$^o$.

The equation $x^2-3xy+\lambda y^2+3x-5y+2=0$,where $\lambda$ is a real number,represents a pair of lines. If $\theta$ is the acute angle between the lines,then $\frac{\operatorname{cosec}^2 \theta}{\sqrt{10}} = $

The angle between the lines $xy = 0$ is equal to .............. $^\circ$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module