So I did the proof and proved their perpendicularity using the Lagrange Multiplier Method, but do not understand the implications of maximizing the distance.
Would the perpendicularity still hold or is this question referring to the gradient vectors?
Since the point $A$ is outside the ellipse and at the point $B$, the pink circle and the ellipse are externally touching, it is the minimum distance. When two curves are touching, it implies they have a common tangent line. Tangent line of a circle is perpendicular to its radius. So, the radius to the common tangent line is minimum.
Similarly, the distance will be maximum when the circle and the ellipse are internally touching. And again the radius to the common tangent line will be maximum.
Example: Optimize $(x-15)^2+(y-10)^2$ subject to $\frac{x^2}{9}+\frac{y^2}{4}=16$.
WA solution. Desmos graph. Note the numbers in Desmos graph are rounded.
Referring to your graph it is:
$\hspace{5cm}$
where: $AB$ is min, $AC$ is max.