Q.If P(x, y) is any point on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \ (a>b)$ whose foci are S and S' then prove that SP + S'P is a constant.

Use the focus-directrix definition: each focal distance equals $e$ times the distance to the corresponding directrix; adding the two focal distances gives the constant $2a$.

Let the ellipse be $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b$), with foci $S=(ae,0)$, $S'=(-ae,0)$ and corresponding directrices $x=\dfrac{a}{e}$, $x=-\dfrac{a}{e}$.

For a point $P(x,y)$ on the ellipse, the focus-directrix property gives:

$SP = e\left(\dfrac{a}{e}-x\right) = a-ex$

$S'P = e\left(x+\dfrac{a}{e}\right) = a+ex$

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