Question

Transform to Cartesian coordinates the equation:

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

Answer 1

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

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

$x=r\cos \theta$

$y=r\sin \theta$

$\cos 2\theta =-{\sin }^2\theta +{\cos }^2\theta$

Using above 3 results in given equation

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

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

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