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

Related to this and this. – Ng Chung Tak Jul 10 '20 at 21:41 — Ng Chung Tak, Jul 10 '20 at 21:41

maxmilgram · Answer 1 · 2020-07-16T10:28:11.063

0

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.

edited Jul 16 '20 at 10:28

answered Jul 10 '20 at 17:21

maxmilgram

3,848
10
14

I did try it but I found the equation of λ in degree 4. I cannot solve this. – Mathskid Jul 16 '20 at 05:23
How to solve it? – Mathskid Jul 16 '20 at 05:23
I am sorry, you are right, I edited the answer. – maxmilgram Jul 16 '20 at 10:25

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

1 Answers1