First slide

Pair of straight lines

Question

If the pair of straight lines $x^{2} - 2 p x y - y^{2} = 0$ and $x^{2} - 2 q x y - y^{2} = 0$ be such that each pair bisects the angle between the other pair, then

Moderate

A

$p q = - 1$
B

$p = q$
C

$p = - q$
D

$p q = 1$

Solution

Equation of the bisector of the angles between $x^{2} - 2 p x y - y^{2} = 0$ is $p x^{2} + 2 x y - p y^{2} = 0$
this same as $x^{2} - 2 q x y - y^{2} = 0$
$∴ \frac{p}{1} = \frac{1}{- q} = \frac{p}{1} \Rightarrow p q = - 1$

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