maximising sum of distances from $(-2,0)$ and $(2,0)$ to the unit circle $x^2+y^2=1$

Question

Consider any point $P$ on the unit circle centred at the origin $O(0,0)$ in $\mathbb R^2$. Let $A$ be $(-2,0)$ and $B$ be $(2,0)$ be two point on the $x$-axis and $D$ be the sum of the two distances $AP$ and $BP$. Then $D$ is maximised when $P$ is at the top (or bottom) of the circle. This is easily proven using calculus.

Can anyone produce a simple direct geometric proof of this result. Calculus seems too heavy for this?

I must admit I was initially attracted to the option where $AP$ is a tangent to the circle, but it is inferior.

For those looking for problems, allowing A and B to be positioned randomly is also quite interesting and specialises to the above.

Answer 1

Let $P=(x,y)$ and $Q=(-x,-y)$ be points on the unit circle. Note that $PA+PB=PA+QA$. $P$ and $Q$ are the end-points of a diameter of the unit circle and $A$ is a point on the circle $x^2+y^2=4$. The problem is equivalent to maximizing $CR+DR$ where $C=(-1,0)$, $D=(1,0)$ and $R$ an arbitrary point on the circle $x^2+y^2=4$. Consider the family of ellipses with foci $C$ and $D$ which meets the circle $x^2+y^2=4$ at some point. The largest ellipse is the one that touches the circle $x^2+y^2=4$ at $(0,2)$ and $(0,-2)$.

Answer 2

Attempt:

1. An ellipse is the locus of points whose sum of distances to two fixed points, the loci, is constant.

2. Consider an ellipse with foci $(-c, 0), (0,c)$, (here: $c=2$), major axis a, minor axis $b$, $b<a$; axes along $X,Y$ axes.

3. Draw a circle centered at $(0,0)$ with radius $1$.

4)The sum of the distances from a point to the foci is $2a$;

and $a^2=c^2+b^2$.

5)Consider the cases:

A) $b >1; $

Then $a^2=c^2+b^2$;

The circle lies within the ellipse.

B) $b=1;$

The circle and ellipse touch at $(0,1)$ and $(0,-1)$.

$a^2=c^2+1=4+1$;

Sum of the lengths $2a=2\sqrt{5};$

C)$b<1$;

The ellipse and circle intersect.

Sum of lengths: $2a=2\sqrt{c^2+b^2}<2\sqrt{5}$.

6)It follows that the maximal sum of lengths $2\sqrt{5}$ is attained for

$b=1$, when circle and ellipse touch.