Using Planetary Motion Laws to relate to elliptic motion

I was reading something to do with elliptic motion and I came across using the equation of orbital velocity to determine the centripetal force acting on a body.

So for an ellipse with a semi-major axis of $a$ and a semi-minor axis of $b$, a planet will experience a force towards the focus of
$$F= frac{GMm}{r^2}$$
and a orbital velocity of
$$ v^2 = GMleft(frac{2}{r} – frac{1}{a}right)$$

And apparently the force obtained would then be $$F= frac{mv^2}{a}$$

I’m not too sure how this was obtained. I’ve tried substituting for $GM$ but I’ve had little success. Perhaps there is another way of approaching this?