Parabola in Complex Form

The parabola y²=4ax in complex form: distance from focus (a+0i) equals distance from directrix Re(z)=−a. The RHS is exactly the distance from z to the directrix. Parametric point: z = at²+2ati = a(t+i)²+... more cleanly: z = at²+2ati. Tangent at parameter t: the line z·t = at²+... or tz = at² gives the tangent form matching the Cartesian ty = x+at². Connects back to: Parabola chapter.

The parabola is defined by a distance equality. In complex form, this definition is stated directly — no equation to derive, just the definition restated.

Focus-Directrix Definition in Complex Language

For $y^2 = 4ax$: focus at $F = a$ (real), directrix $\operatorname{Re}(z) = -a$.

The distance from $z = x + iy$ to the focus:

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

The distance from $z$ to the directrix $x = -a$ is:

$$x + a = \operatorname{Re}(z) + a$$

The parabola is the locus where these are equal:

$$|z - a| = \operatorname{Re}(z) + a$$

Verification: Squaring both sides (the right side is non-negative for points on or to the right of the directrix):

$$(x-a)^2 + y^2 = (x+a)^2$$ $$x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2$$ $$y^2 = 4ax \qquad \checkmark$$

Parametric Point

The parametric form of $y^2 = 4ax$ is $(at^2, 2at)$, $t \in \mathbb{R}$. As a complex number:

$$z = at^2 + 2iat = at^2 + i(2at)$$ $$z = a(t^2 + 2it)$$

Tangent at Parameter $t$

The Cartesian tangent at $(at^2, 2at)$ is $ty = x + at^2$.

In complex form, substituting $x = \operatorname{Re}(z)$ and $y = \operatorname{Im}(z)$:

$$t \cdot \operatorname{Im}(z) = \operatorname{Re}(z) + at^2$$

Using $\operatorname{Re}(z) = (z+\bar{z})/2$ and $\operatorname{Im}(z) = (z-\bar{z})/(2i)$:

$$\frac{t(z - \bar{z})}{2i} = \frac{z + \bar{z}}{2} + at^2$$ $$t(z - \bar{z}) = i(z + \bar{z}) + 2iat^2$$ $$z(t - i) = \bar{z}(t + i) + 2iat^2$$

This is the tangent in complex form — less clean than the Cartesian version for computation, but it connects to the general line form $\bar{a}z + a\bar{z} + b = 0$.

Focal Chord Condition

For a focal chord (passes through focus $a$), the two parameters $t_1$ and $t_2$ satisfy $t_1 t_2 = -1$. In complex form: the two parametric points are $a(t^2 + 2it)$ and $a(1/t^2 - 2i/t)$ — negatives in the imaginary part, consistent with the chord passing through the real point $a$.