Question

Transform to Cartesian coordinates the equation:

$r={\alpha }^2$

Answer 1

In polar co-ordinates,

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

$x=r\cos \theta$

$y=r\sin \theta$

$\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

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

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

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

Squaring both sides, we get

$x^2+y^2={\alpha }^4$