Normal to an ellipse

Question

A normal is drawn to the ellipse $\frac{x^2}{(a^2+2a+2)^2}+\frac{y^2}{(a^2+1)^2}=1$. If maximum radius of the circle centered at the origin and touching the normal is $5$, then find the possible values of '$a$'.

Answer 1

Hints:

This ellipse has valid values of $a$ in the positive and negative real numbers.

The equation of the circle centered at the origin and having maximum radius $5$ is where $r\in[0,5]$ and

$$x^2+y^2=r^2\implies \frac {x^2}{r^2}+\frac{y^2}{r^2}=1$$

Which direction does a "normal to an ellipse" travel in, relative to the center of the ellipse?