Let the point P be (acosθ,bsinθ)
So, the polar line w.r.t hyperbola will be:
xacosθa2−ybsinθb2=1
Or, xcosθa−ysinθb=1
The asymptotes of the hyperbola is:
y=±bxa
The point of intersection Q will be:
xcosθa−bxsinθab=1
Or, x=acosθ−sinθ and y=bcosθ−sinθ
And, the point of intersection R will be:
xcosθa+bxsinθab=1
Or, x=acosθ+sinθ and y=−bcosθ+sinθ
Given, QR=c Or a2(cosθ+sinθ−cosθ+sinθ)2((cosθ)2−(sinθ)2)+b2(cosθ+sinθ+cosθ−sinθ)2((cosθ)2−(sinθ)2)=c2
Or 4a2(sinθ)2+4b2(cosθ)2=c2(cos2θ)2
Or 4b2+4(a2−b2)(sinθ)2=c2(cos2θ)2
Or −4b2−4(a2−b2)(sinθ)2=−c2(cos2θ)2
Adding and subtracting 2(a2−b2), we get
−4b2−2(a2−b2)+2(a2−b2)(1−2(sinθ)2)=−c2(cos2θ)2
2(−a2−b2)+2(a2−b2)(cos2θ)=−c2(cos2θ)2
c2(cos2θ)2+2(a2−b2)(cos2θ)=2(a2+b2)