Question

Transform to Cartesian coordinates the equation:

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

Answer 1

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

Squaring the given equation, we get

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

$2{\sin }^2\frac{\theta }{2}=1-\cos \theta$

Using this result in above equation

$\Rightarrow r=\alpha (1-\cos \theta )\times \frac{1}{2}$

$\Rightarrow 2r=\alpha (1-\frac{x}{r})$ ...... $[\because x=r\cos \theta ]$

$\Rightarrow 2r^2=\alpha (r-x)$

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

$\Rightarrow 2(x^2+y^2)+\alpha x=\alpha r$

Squaring both sides

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

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