Ellipse as a Complex Locus

Locus of z whose sum of distances from two fixed points z₁ (focus 1) and z₂ (focus 2) is constant 2a. This is the definition of an ellipse — stated most cleanly in complex form. The condition 2a > |z₁−z₂| ensures the locus is non-degenerate.

The ellipse has two equivalent definitions — one Cartesian, one as a locus of distances. The complex form states the locus definition directly, and the Cartesian form follows by algebra.

The Locus Condition

Let $z_1$ and $z_2$ be the two foci, and let $2c = |z_1 - z_2|$ be the distance between them. The locus:

$$|z - z_1| + |z - z_2| = 2a, \qquad a > c$$

is the set of all points whose sum of distances from the two foci is constant.

Reducing to Cartesian Form

Place the foci symmetrically on the real axis: $z_1 = c$, $z_2 = -c$. Let $z = x + iy$.

$$|z - c| + |z + c| = 2a$$ $$\sqrt{(x-c)^2 + y^2} + \sqrt{(x+c)^2 + y^2} = 2a$$

Move one radical to the right and square:

$$\sqrt{(x-c)^2 + y^2} = 2a - \sqrt{(x+c)^2 + y^2}$$ $$(x-c)^2 + y^2 = 4a^2 - 4a\sqrt{(x+c)^2+y^2} + (x+c)^2 + y^2$$

Simplifying the left and right:

$$-4cx - 4a^2 = -4a\sqrt{(x+c)^2+y^2}$$ $$a^2 + cx = a\sqrt{(x+c)^2+y^2}$$

Squaring again:

$$a^4 + 2a^2cx + c^2x^2 = a^2(x^2 + 2cx + c^2 + y^2)$$ $$a^4 + c^2x^2 = a^2 x^2 + a^2 c^2 + a^2 y^2$$ $$(a^2 - c^2)x^2 + a^2 y^2 = a^2(a^2 - c^2)$$

Setting $b^2 = a^2 - c^2$ (with $b > 0$ since $a > c$):

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

This is the standard Cartesian ellipse with semi-major axis $a$ and semi-minor axis $b$. $\blacksquare$

The Non-Degeneracy Condition

The condition $2a > |z_1 - z_2|$ (equivalently $a > c$) ensures $b^2 = a^2 - c^2 > 0$, giving a genuine ellipse. When $2a = |z_1 - z_2|$, the locus degenerates to the line segment joining $z_1$ and $z_2$.

Why the Complex Form Is Cleaner

The Cartesian derivation requires two rounds of squaring and careful bookkeeping. The complex form $|z - z_1| + |z - z_2| = 2a$ states the geometric truth in one line — distance from one focus plus distance from the other focus equals constant. No coordinates, no axis placement required.