newtonian mechanics - Satellite in Elliptical orbit

When finding the period of a satellite orbiting the earth we equate the centripetal force to the gravitational force

$$\frac{mv^2}{r} = \frac{-GMm}{r^2}$$ If I understood well the $r$ cancels into the $r^2$ because the distance from the earth at any point on the orbit equals the radius of the orbit. Now what if we have an ellipse instead of a circle?

And by the way why does Kepler's $1^{st}$ law say In elliptical orbits