Parabola: Focus, Directrix, and Equidistance Property

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This elegant geometric definition produces the familiar U-shaped curve.

Standard form

For a parabola opening upward with vertex at the origin:

$y = \dfrac{1}{4p}x^{2}$

or equivalently $x^{2} = 4py$, where:

• Focus is at $(0, p)$
• Directrix is the line $y = -p$
• $p$ is the distance from vertex to focus (and vertex to directrix)

For $y = x^{2}$: comparing with $y = \frac{1}{4p}x^{2}$, we get $\frac{1}{4p} = 1$, so $p = 1/4$.

• Focus: $(0, 1/4)$
• Directrix: $y = -1/4$

Verifying the equidistance property

Take any point $(x, x^{2})$ on $y = x^{2}$:

Distance to focus $(0, 1/4)$: $d_{F} = \sqrt{x^{2} + (x^{2} - 1/4)^{2}} = \sqrt{x^{2} + x^{4} - x^{2}/2 + 1/16}$

Distance to directrix $y = -1/4$: $d_{D} = x^{2} + 1/4$

With algebra, $d_{F} = \sqrt{(x^{2} + 1/4)^{2}} = x^{2} + 1/4 = d_{D}$. They're equal for every point! ✓

Real-world parabolas

• Satellite dishes: Parallel signals from a satellite reflect off the parabolic dish and converge at the focus, where the receiver sits.
• Car headlights: A bulb at the focus of a parabolic reflector produces a parallel beam — the reverse of the satellite dish.
• Bridges: Suspension bridge cables hang in a parabolic shape (approximately; technically a catenary under self-weight alone, but a parabola under uniform horizontal load).

Try it in the visualization

Adjust $p$ to change the parabola's width and focus position. Toggle "equidistance" to see distance lines from a moving point to both the focus and the directrix — they're always equal. Change the orientation (up/down/left/right) to see all four parabola types.