Question

find the equation of the system of orthogonal trajectories of the family of curves r^n sin n(theta)=a^n where a is the parameter​

Answer 1

Correct option is

C

r

8

=c

8

sin4θ

We have, r

2

=a

2

cos4θ=a

2

(1−2sin

2

2θ)...(1)

Differentiating w.r.t. θ, we get

2r

dθ

dr

=−4a

2

sin4θ...(2)

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

r

2

dθ

dr

=−

cos4θ

4sin4θ

...(3)

Replacing

dθ

dr

with −r

2

dr

dθ

in (3), we get

2r

dr

dθ

=

cosθ

4sin4θ

⟹

r

2

dr=

sinθ

cosθ

dθ

Integrating, we get

2logr=

4

1

logsin4θ+2logc

⟹r

8

=c

8

sin4θ

Answer 2

$\sec \: n \: theta \: + r = c$