The pair of lines l x^2 + 2(l+m) x y + m y^2 | vedclass

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(C) Let the angle between the given pair of lines be $	heta$.According to the given information,the area of the bigger sector is $5$ times the area o

Vedclass · Accepted Answer

$\frac{11}{12}$

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(C) Let the angle between the given pair of lines be $	heta$.According to the given information,the area of the bigger sector is $5$ times the area of the smaller sector.Let $A_1$ be the area of the smaller sector and $A_2$ be the area of the bigger sector.$A_2 = 5 A_1 \Rightarrow \frac{1}{2}(\pi - 	heta) r^2 = 5 	imes \left(\frac{1}{2} 	heta r^2ight)$$\Rightarrow \pi - 	heta = 5 	heta$ $\Rightarrow 6 	heta = \pi$ $\Rightarrow 	heta = \frac{\pi}{6}$The angle $	heta$ between the pair of lines $a x^2 + 2h x y + b y^2 = 0$ is given by $	an 	heta = \left| \frac{2 \sqrt{h^2 - ab}}{a + b} ight|$.Here,$a = l$,$h = l+m$,and $b = m$.$	an \frac{\pi}{6} = \frac{1}{\sqrt{3}} = \left| \frac{2 \sqrt{(l+m)^2 - lm}}{l+m} ight|$Squaring both sides:$\frac{1}{3} = \frac{4((l+m)^2 - lm)}{(l+m)^2}$$(l+m)^2 = 12(l+m)^2 - 12 lm$$11(l+m)^2 = 12 lm$$\frac{lm}{(l+m)^2} = \frac{11}{12}$Hence,option $(c)$ is correct.

The pair of lines $l x^2 + 2(l+m) x y + m y^2 = 0$ lies along two diameters of a circle and divides the circle into $4$ sectors. If the area of the bigger sector is $5$ times the area of the smaller sector,then $\frac{l m}{(l+m)^2} = $

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