If the angle between the lines represented by | vedclass

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

Question

(A) The given equation of the pair of lines is $x^2+\lambda xy-y^2 	an^2 	heta=0$.Comparing this with the general form $ax^2+2hxy+by^2=0$,we get $a=

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) The given equation of the pair of lines is $x^2+\lambda xy-y^2 	an^2 	heta=0$.Comparing this with the general form $ax^2+2hxy+by^2=0$,we get $a=1$,$h=\frac{\lambda}{2}$,and $b=-	an^2 	heta$.The angle $\alpha$ between the pair of lines is given by $	an \alpha = \left|\frac{2\sqrt{h^2-ab}}{a+b}ight|$.Here,$\alpha = 2	heta$,so $	an 2	heta = \left|\frac{2\sqrt{(\frac{\lambda}{2})^2 - (1)(-	an^2 	heta)}}{1-	an^2 	heta}ight|$.Using the identity $	an 2	heta = \frac{2	an 	heta}{1-	an^2 	heta}$,we have $\frac{2	an 	heta}{1-	an^2 	heta} = \left|\frac{2\sqrt{\frac{\lambda^2}{4}+	an^2 	heta}}{1-	an^2 	heta}ight|$.Squaring both sides,we get $\frac{4	an^2 	heta}{(1-	an^2 	heta)^2} = \frac{4(\frac{\lambda^2}{4}+	an^2 	heta)}{(1-	an^2 	heta)^2}$.This simplifies to $	an^2 	heta = \frac{\lambda^2}{4} + 	an^2 	heta$,which implies $\frac{\lambda^2}{4} = 0$.Therefore,$\lambda = 0$.

If the angle between the lines represented by the equation $x^2+\lambda xy-y^2 \tan^2 \theta=0$ is $2\theta$,then the value of $\lambda$ is

Similar Questions

If $\theta$ is the acute angle between the lines represented by the equation $2x^2 + 7xy + 3y^2 = 0$, then the value of $\frac{2 \cos \theta - 3 \sin \theta}{4 \sin \theta + 5 \cos \theta}$ is:

The acute angle between the lines $(x^2+y^2) \sin \theta+2xy=0$ is

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}} = $

If the pair of lines $ax^2 + 2(a + b)xy + by^2 = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector,then:

If the slope of one of the lines represented by $5x^2 + \frac{40}{3}xy + ky^2 = 0$ is $3$,then the angle between the pair of lines is

Vedclass Test Series

Exam Paper Generator

Online Exam Module