Geo3.12

If the pair of line $ax^2+2pxy-ay^2=0,bx^2+2qxy-by^2=0\,$ are such that each pair bisects the angle between the other pair,then show that $ab+pq=0\,$

Equation to the pair of angle bisectors of $ax^2+2pxy-ay^2=0\,$

$\frac{x^2-y^2}{a+a}=\frac{xy}{p}\,$

$px^2-2axy-py^2=0\,$

But the angle bisectors are $bx^2+2qxy-by^2=0\,$

$\frac{p}{b}=\frac{-2a}{2q}=\frac{-p}{-b}\,$

$pq=-ab\,$

$pq+ab=0\,$