Question

Transform to Cartesian coordinates the equation:

$r^2={\alpha }^2\cos 2\theta$

Answer 1

Given equation is

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

We know that, $\cos 2\theta =-{\sin }^2\theta +{\cos }^2\theta$

Substituting in $\left(i\right)$, we get

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

$x=r\cos \theta$

$y=r\sin \theta$

Using above result

$\Rightarrow r^2={\alpha }^2(\frac{x^2}{r^2}-\frac{y^2}{r^2})$

$\Rightarrow r^4={\alpha }^2(x^2-y^2)$

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

$\Rightarrow (x^2+y^2{)}^2={\alpha }^2(x^2-y^2)$