Question

The orthogonal trajectories for the family of curves $r^2=a^2\cos 4\theta$ are:

• A. $r^2=c^2\sin \theta$
• B. $r^4=c^4\sin 2\theta$
• C. $r^8=c^8\sin 4\theta$
• D. $r^8=c^8\sin 2\theta$

Answer 1

The correct option is C $r^8=c^8\sin 4\theta$

We have, $r^2=a^2\cos 4\theta =a^2(1-2{\sin }^{2}2\theta )...\left(1\right)$

Differentiating w.r.t. $\theta$, we get

$2r\frac{dr}{d\theta }=-4a^2\sin 4\theta ...\left(2\right)$

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

$\frac{2}{r}\frac{dr}{d\theta }=-\frac{4\sin 4\theta }{\cos 4\theta }...\left(3\right)$

Replacing $\frac{dr}{d\theta }$ with $-r^2\frac{d\theta }{dr}$ in (3), we get

$2r\frac{d\theta }{dr}=\frac{4\sin 4\theta }{\cos \theta }$

$⟹\frac{2}{r}dr=\frac{\cos \theta }{\sin \theta }d\theta$

Integrating, we get

$2\log r=\frac{1}{4}\log \sin 4\theta +2\log c$

$⟹r^8=c^8\sin 4\theta$