Find shortest distance from point on ellipse to focus of ellipse.

Since focal distance $ = e \times$distance from directrix ($e=$eccentricity = $\frac{\sqrt{a^2-b^2}}{a}$), we only need to locate points on the ellipse that are closest to a directrix and those would be the corresponding end point of the major axis in this case $(a,0)$

Assuming $a>b$, we have $c=\sqrt{a^2-b^2}$.

Using the parametric equations

$$\begin{cases}x=a\cos\theta,\\y=b\sin\theta\end{cases}$$

we minimize

$$(a\cos\theta-c)^2+(b\sin\theta)^2.$$

The derivative of this expression is

$$-a\sin\theta(a\cos\theta-c)+b\cos\theta(b\sin\theta).$$

There is a root for $$\sin\theta=0$$ (on the major axis) and for

$$(a^2-b^2)\cos\theta=ac,$$ or

$$\cos\theta=\frac a{\sqrt{a^2-b^2}}>1 (!).$$

Hence,

$$d=a-\sqrt{a^2-b^2}.$$