Sum of Focal Distances

The defining property of an ellipse: the sum of distances from any point on the ellipse to the two foci is constant, equal to the major axis length 2a.

From the focal radii: $r_1 = a - ex_1$ and $r_2 = a + ex_1$.

$$r_1 + r_2 = (a - ex_1) + (a + ex_1) = 2a$$

The sum is independent of $x_1$ — it is the same for every point on the ellipse. This constant equals $2a$, the length of the major axis.

This is both the definition and a theorem:

• As a definition: an ellipse is the locus where $r_1 + r_2 = 2a$ (constant $> 0$, with two fixed foci).
• As a theorem: any point satisfying the Cartesian equation $x^2/a^2 + y^2/b^2 = 1$ satisfies $r_1 + r_2 = 2a$.

Practical construction: Fix two pins (the foci), loop a string of length $2a$ around them, and draw taut with a pencil. The pencil traces an ellipse.

Range of focal radii: Since $-a \leq x_1 \leq a$:

$$a(1-e) \leq r_1 \leq a(1+e), \quad a(1-e) \leq r_2 \leq a(1+e)$$

Minimum focal distance (at vertex nearer to focus): $a(1-e) = a - c$. Maximum (at vertex farther from focus): $a(1+e) = a + c$.