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Q.

If the pairs of lines $3 x^{2} - 2 p x y - 3 y^{2} = 0 and 5 x^{2} - 2 q x y - 5 y^{2} = 0$ are such that each pair bisects the
angle between the other pair, then pq equals to _______

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answer is -15.

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Detailed Solution

The equation of bisectors of $3 x^{2} - 2 p x y - 3 y^{2} = 0$ is

$\frac{x^{2} - y^{2}}{3 - (- 3)} = \frac{x y}{- p}$ $\Rightarrow p x^{2} + 6 x y - p y^{2} = 0$

But this is same as $5 x^{2} - 2 q x y - 5 y^{2} = 0$

$∴ \frac{p}{5} = \frac{6}{- 2 q} \Rightarrow p q = - 15$

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