If S and S' are the foci of the ellipse x^218+y^29=1 and P is a point on the ellipse, then…

Question

If $S$ and $S'$ are the foci of the ellipse $\dfrac{x^2}{18}+\dfrac{y^2}{9}=1$ and $P$ is a point on the ellipse, then $\min(SP\cdot S'P)+\max(SP\cdot S'P)$ is equal to:

Accepted Answer

$a=3\sqrt2,\ b=3,\ c=3,\ e=\dfrac{1}{\sqrt2}$. $SP+S'P=2a=6\sqrt2$.

By AM--GM, $SP\cdot S'P\le\left(\dfrac{SP+S'P}{2}\right)^2=18$ (maximum, at the end of the minor axis).

At a vertex on the major axis, $SP=a-c=3\sqrt2-3,\ S'P=a+c=3\sqrt2+3$, so $SP\cdot S'P=18-9=9$ (minimum).

$$\min+\max = 9+18=27.$$

Solutions · 1