Calculate the maximum distance from a given point to a point on an eclipse

Question

I have a problem as below: Given a complex number $z$ which satisfies the expression: $$ |z - 10| + |z + 10| = 12 \sqrt5 $$

Find the maximum value for $$ P = |z - 4 - 22i| $$

For this problem, I have a prediction as follow:

as you can see in the picture, that is the intuitive graph for the eclipse of the given expression and the point $A(4, 22)$ for the expression $P$

I predict that the value of $P$ is maximized if it is on the green line and crossed with the eclipse. Therefore the maximum value of $P$ is from the point $A$ to the crossed point of the green line with the eclipse.

However, that is my prediction for this problem without proof. Luckily, it gives the right answer but I have no idea for any reason.

A straight-forward approach would be to parametrize the ellipse (on the form $a \cos (\theta) + i b \sin(\theta)$ ), find an expression for the distance and optimize it wrt $\theta$. — Thusle Gadelankz, Apr 11 '21 at 06:05
Posted also on [mathoverflow.se]: Calculate the maximum distance from a given point to a point on an eclipse. On [math.meta.se] you can find several posts which give some advice concerning cross-posting. — Martin Sleziak, Apr 11 '21 at 06:08
It reduces by translation to find a point on ellipse closest to origin, where answers address both extrema. — dxiv, Apr 11 '21 at 07:32

score 1 · Answer 1 · answered Apr 11 '21 at 07:12

1

If the point of intersection $p=(-5.2\dots,-8.2\dots)$ maximised $P$, the green line would be orthogonal to the ellipse at $p$, i.e. the gradient of the ellipse's implicit representation $\frac{x^2}{180}+\frac{y^2}{80}=1$ at $p$ would be in the same direction as the green line. It is not: the green line's slope is $\frac{22-4\sqrt5}4=3.263\dots$ while the gradient is $3.518\dots$

The correct maximiser is obtained by obtaining the equation corresponding to green line orthogonality and solving it. This gives $p=(-6,-8)$ for a maximal $P$ of $10\sqrt{10}$.

answered Apr 11 '21 at 07:12

Parcly Taxel

103,344

Can you explain why if the green line is orthogonal to the ellipse at p, the distance will be maximised? – Hoang Nam Apr 11 '21 at 08:17
@HoangNam The gradient of an implicit function is the direction of steepest change. If the green line was not orthogonal, we could move on the ellipse and get a higher value for $P$. – Parcly Taxel Apr 11 '21 at 12:45

score 0 · Answer 2 · answered Apr 11 '21 at 07:31

Also, another interesting observation of the answer is obtained by replacing $w=z-22i-4$. Hence the optimization problem is $$ {\max |w| \\ \text{s.t.} \\ \ \ \ |w+22i-6|+|w+22i+14|=12\sqrt5 } $$with the following visualization , where a trivial answer does not seem trivial (I have sketched the outer circle numerically and inaccurately) and I suggest a bit of calculus, as @ParclyTaxel mentioned.

Calculate the maximum distance from a given point to a point on an eclipse

2 Answers2