Question

Transform to Cartesian coordinates the equation:

$r^{\frac{1}{2}}\cos \frac{\theta }{2}={\alpha }^{\frac{1}{2}}$

Answer 1

$r^{\frac{1}{2}}\cos \frac{\theta }{2}={\alpha }^{\frac{1}{2}}$ ....... $\left(i\right)$

Squaring the given equation, we get

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

But $2{\cos }^2\frac{\theta }{2}=1+\cos \theta$

$\therefore r(1+\cos \theta )=2\alpha$

Also, $x=r\cos \theta$ and $y=r\sin \theta$

$\Rightarrow r(1+\frac{x}{r})=2\alpha$

$\Rightarrow r+x=2\alpha$

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

Squaring both sides

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

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