Question

Change to polar coordinates the equation: $(x^2+y^2{)}^2=a^2(x^2-y^2)$

Answer 1

Given equation is

$(x^2+y^2{)}^2=(\alpha {)}^2(x^2-y^2)$ .... $\left(i\right)$

In polar co-ordinates,

$r=\sqrt{x^2+y^2}$

$x=r\cos \theta$

$y=r\sin \theta$

Also, we know that, $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$

Using above results in equation $\left(i\right)$, we get

$(r^2{)}^2={\alpha }^2(r^2{\cos }^2\theta -r^2{\sin }^2\theta )$

$\Rightarrow r^2={\alpha }^2({\cos }^2\theta -{\sin }^2\theta )$

$\Rightarrow r^2={\alpha }^2\cos \left(2\theta \right)$