Converting polar equation to cartesian coordinates

Question

I have been trying everything to convert the polar equation $r=\frac{2}{1-\cos(\theta)}$ to cartesian coordinates but I simply didn't manage to know the right answer. Please help me..

Answer 1

Let's do some manipulation on your equation. $$r=\frac{2}{1-\cos(\theta)}$$ $$r-r/cos(\theta)=2$$

Now we can convert with $r=\sqrt{x^2+y^2}$ and $x=r\cos(\theta)$ $$\sqrt{x^2+y^2}-x=2$$ $$\sqrt{x^2+y^2}=2+x$$ $$x^2+y^2=x^2+4x+4 $$ $$y^2=4x+4 $$ So we get $$y=\pm2\sqrt{x+1}$$

Answer 2

Multiply both sides by $1- cos(\theta)$: $r(1- cos(\theta))= r- rcos(\theta)= 2$. Replace r with $\sqrt{x^2+ y^2}$ and $rcos(\theta)$ by x: $\sqrt{x^2+ y^2}- x= 2$. Then $\sqrt{x^2+ y^2}= x+ 2$ so that, squaring both sides, $x^2+ y^2= (x+ 2)^2= x^2+ 4x+ 4$. The $x^2$ terms cancel so $y^2= 4x+ 4$.