How to find the general form of minimum distance from the point (m,n) to the ellipse by using Lagrange Multiplier?

Question

Excuse me!

I have tried to solve this problem for a long time but I have stuck in this step.

This is my work picture01.

This is my work picture02.

I cannot simplify y in term of a b m and n. It’s too complicated.

Could you guide me how to do this? And Do you have any method to solve this equation?

Thank you very much

Answer 1

You solve the first equation for $x$ and the second equation for $y$. Then you substitute both in the third equation. Unfortunately this results in a quartic equation in $\lambda$ which can be solved by Ferrari's method.

I don't think there is way around the forth order equation since depending on the position of the point $(m,n)$ there are up to four critical points.