Minimum and Maximum Distance to an Ellipse

You are given the ellipse

$\large \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

with $a = 2$ and $b = 1$ . Determine the minimum $m$ and the maximum $M$ distance from the point $(5,3)$ to the ellipse, as shown in the above figure.

Submit the difference $M - m$ .

Hint: Use Lagrange multiplier method.