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Question

If a pair of lines x22pxyy2=0 and x22pxyy2=0 is such that each pair bisects the angle between the other pair, prove that pq=1.

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Solution

Equations of the bisectors of x22pxyy2=0

are x2y21(1)=xyp

or x2y2=2xyp ..(1)

But the bisectors of first pair of lines are given by the 2nd paid, i.e., by
x2y2=2qxy ..(2)

Comparing (1) and (2), we get

1/p=q or pq=1

or pq+1=0.


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