Parabola Opening Right

Vertex at origin, axis along positive x-axis. Focus at (a, 0), directrix x = −a, latus rectum x = a.

A parabola is the locus of all points equidistant from a fixed point $S$ (the focus) and a fixed line $\ell$ (the directrix), with the focus not lying on the directrix.

Setup: Place the vertex at the origin. Let the focus be $S(a, 0)$ and directrix $x = -a$ (so the axis of symmetry is the x-axis).

Let $P(x, y)$ be any point on the parabola. Let $M$ be the foot of the perpendicular from $P$ to the directrix.

$$PM = x + a \quad \text{(horizontal distance from } P \text{ to } x = -a\text{)}$$ $$PS = \sqrt{(x-a)^2 + y^2}$$

By definition $PS = PM$:

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

Squaring (valid since both sides are non-negative when $x \geq 0$, and $x + a > 0$ for the right-opening case):

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

Summary of elements for $y^2 = 4ax$: