Question

show that the family of curves r^n= a sec n theta and r^n=b cosec n theta are orthogonal
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Answer 1

We have, r2=a2cos4θ=a2(1−2sin22θ)...(1)

Differentiating w.r.t. θ, we get

2rdθdr=−4a2sin4θ...(2)

Eliminating a from (2) using (1), we get

r2dθdr=−cos4θ4sin4θ...(3)

Replacing dθdr with −r2drdθ in (3), we get

2rdrdθ=cosθ4sin4θ

⟹r2dr=sinθcosθdθ

Integrating, we get

2logr=41logsin4θ+2logc

⟹r8=c8sin4θ

I hope it will help you.

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Answer 2

Answer:

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