Equation Conversion: Polar to Rectangular

Question

Convert the polar equation to rectangular form (rectangular equation)

$$r=\frac{9}{1-3\cos(\theta)}$$

I know that $r^2= x^2+y^2, x= r\cos(\theta)$ and $y= r\sin(\theta)$ and $\tan(\theta)= y/x$

I don't even understand how to get started.

Answer 1

Assuming $$r=\frac{9}{1-3\cos(\theta)}$$ then we can multiply by $1-3\cos(\theta)$ to get $$r-3r\cos(\theta)=9$$ now substituting $r=\sqrt{x^2+y^2}$ and $r\cos(\theta)=x$ to get $$\sqrt{x^2+y^2}-3x=9$$ $$\sqrt{x^2+y^2}=9+3x=3(3+x)$$ square both sides

$$x^2+y^2=3^2(3+x)^2=9(x^2+6x+9)$$ $$y^2=8x^2+54x+81$$ thus we gain that $$y=\pm \sqrt{8x^2+54x+81}$$