If p and q are the lengths of perpendiculars from the origin to the lines xcosθ -ysinθ =kcos2θ and xsecθ +ycosθ =k, respectively, prove that p²+4q²=k²

Given lines are xcosθ -ysinθ = -kcos2θ

And xsecθ +ycosecθ =k

Now, we find perpendicular of these lines from the origin

P = and q=

P=kcos2θ and q = kcosθ sinθ

P=kcos2θ and q =

Cos2θ = , q= i.e. sin2θ =

Since cos²2θ +sin²2θ =1

We have,

Or p²+4q²=k²