$$r(\phi)=\frac{c}{1+\epsilon\cos\phi}$$

Prove that this is the equation of an ellipse for $0\le\epsilon<1$ by converting to Cartesian coordinates.

$\displaystyle x=r\cos\phi=\frac{c\cos\phi}{1+\epsilon\cos\phi}$

$\displaystyle y=r\sin\phi=\frac{c\sin\phi}{1+\epsilon\cos\phi}$

How do I eliminate $\phi$? Any suggestions?