Question

Transform to Cartesian coordinates the equation:

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

Answer 1

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

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

$x=r\cos \theta$.... $\left(ii\right)$

$y=r\sin \theta$.... $\left(iii\right)$

$\theta ={\tan }^{-1}\frac{y}{x}$

Substituting the identity in above equation, we get

$\sin 2\theta =2\sin \theta \cos \theta$

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

Substituting $\left(ii\right)$ and $\left(iii\right)$, we get

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

$\Rightarrow xy={\alpha }^2$