If the pair of straight lines `x^(2)-2pxy-y^(2)=0and x^(2)-2qxy-y^(2)=0` are such that each pair bisects the angle between the other pair , then prove that `pq=-1`.

Explanation

Combined equation of the bisectors of the angles between pair of lines `x^(2)-2pxy-y^(2)=0` is
`(x^(2)-y^(2))/(1-(-1))=(xy)/(-p)`
or `px^(2)+2xy-py^(2)=0`
This must be same as the given equation of pair of bisectors i.e., `x^(2)-2qxy-y^(2)=0`.
Comparing ratios of coefficients , we get
`(p)/(1)=(1)/(-q)=(p)/(1)`
`:.pq=-1`