Question

if x= p tan theta +q sec theta and y=p sec theta q tan theta then prove that x²-y²=q²-p²​

Answer 1

x=ptanθ+qsecθ and y=psecθ+qtanθ

x²=p²tan²θ+2pqtanθsecθ+q²sec²θ

and

y²=p²sec²θ+2pqsecθtanθ+q²tan²θ

∴, x²-y²

=p²tan²θ+2pqtanθsecθ+q²sec²θ-p²sec²θ-2pqtanθsecθ-q²tan²θ

=tan²θ(p²-q²)-sec²θ(p²-q²)

=(p²-q²)(tan²θ-sec²θ)

=(p²-q²).(-1) [∵, tan²θ-sec²θ=-1]

=q²-p² (Proved)