Question

Find the equivalent cartesian equation $\sqrt{r}=\sqrt{a}\cos \frac{\theta }{2}$.

Answer 1

Given,

$\sqrt{r}=\sqrt{a}\cos \frac{\theta }{2}$

or, $r=a{\cos }^2\frac{\theta }{2}$

or, $2r=a(1+\cos \theta )$ $[\because 1+\cos \theta =2{\cos }^2\frac{\theta }{2}]$

or, $2r^2=ar+a\left(r\cos \theta \right)$

or, $(2r^2-ax)=arr[\because x=r\cos \theta ]$

or, $(2x^2+2y^2-a{x}^2)=a^2(x^2+y^2)$

[Squaring both sides we get] and $r=\sqrt{x^2+y^2}$.