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Q.

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 bigger sector is 5 times the area of smaller sector, then $\frac{l m}{{(l + m)}^{2}} =$

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a

$\frac{1}{2}$

b

$\frac{2}{\sqrt{3}}$

c

$\frac{11}{12}$

d

$\frac{13}{12}$

answer is C.

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Detailed Solution

$θ$ is acute angle between pair of lines $l x^{2} + 2 (l + m) x y + m y^{2} = 0$ area of big sector = 5 area of small sector $r^{2} (π - θ) = 5 r^{2} θ$
$\cos π / 6 = | \frac{l + π}{{(l - m)}^{2} + 4 {(l + m)}^{2}} |$ $\frac{l m}{{(l + m)}^{2}} = \frac{11}{12}$

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