The Focal-Distance Property

The sum-of-distances theorem

The most famous fact about an ellipse — the one behind the classic "pin-and-string" construction — is this:

Theorem. If $P(x,y)$ is any point on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b$) with foci $S(ae,0)$ and $S'(-ae,0)$, then $SP + S'P$ is the same constant for every point $P$ on the curve, namely $2a$ — the length of the major axis.

Why it's true. Each focus comes with its own directrix, and the defining rule $SP = e\cdot(\text{distance to that directrix})$ applies to both pairs simultaneously. Writing $Z, Z'$ for the feet of the perpendiculars from $C$ to the two directrices ($CZ = a/e$, $CZ'=a/e$), and $L$ for the foot of the perpendicular from $P$ to the vertical line through $S$:

Adding these, the $ex$ terms cancel exactly:

The individual focal distances $SP = a-ex$ and $S'P = a+ex$ are worth remembering on their own — they let you find the distance from either focus to any point on the ellipse using only its x-coordinate, without any square roots.

Why this gives a physical way to draw an ellipse

Because $SP+S'P=2a$ is constant, you can draw an ellipse mechanically: fix two pins at the foci, loop a piece of string of length $2a$ around them, and trace the curve keeping the string taut with a pencil. This works precisely because the two focal distances always add to the same total — it is also why an ellipse is sometimes defined as "the locus of a point whose distances from two fixed points add up to a constant" (equivalent to the focus-directrix definition, provided that constant exceeds the distance between the two fixed points).

Worked Example

Problem. For the ellipse $\dfrac{x^2}{100} + \dfrac{y^2}{64} = 1$, find the sum of the focal distances of any point on the ellipse, and then find the individual distances $SP$ and $S'P$ for the point $P(6, \tfrac{32}{5})$ on the ellipse. …