Question

Convert the rectangular equation to polar form

Answer 1

The coordinate change from rectangular to polar is

$\begin{cases}x=\rho\cos(\theta)\\y=\rho\sin(\theta)\end{cases}$

So, you simply have to substitute each occurrence of x and y with their expression:

$(x^2+y^2)^2 = x^2-y^2 \iff (\rho^2\cos^2(\theta)+\rho^2\sin^2(\theta))^2 = \rho^2\cos^2(\theta)-\rho^2\sin^2(\theta)$

Using the fundamental trigonometric equation $\cos(x)^2+\sin(x)^2=1$ we have

$(\rho^2)^2 = \rho^2(\cos^2(\theta)-\sin^2(\theta)) \iff \rho^4-\rho^2(\cos^2(\theta)-\sin^2(\theta))=0$

We can use $\cos(2x)=\cos^2(x)-\sin^2(x)$ to get

$\rho^4-\rho^2(\cos(2\theta))=0 \iff \rho^2(\rho^2-\cos(2\theta))=0$