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Derive the equation of a parabola from a focus and directrix.

Derive separate parabola features from focus and directrix

Problem
For a parabola with focus \((0,~2)\) and directrix \(y~=~-2\), project the focus perpendicularly to the directrix and take the midpoint for the vertex. Report the axis, signed focal parameter \(p\), opening direction, and equal-distance check.
Coordinate plane with focus (0,2) and horizontal directrix y=−2; vertex, axis, p-value, and parabola are not shown. Open full size
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Hint

Identify whether the directrix is horizontal or vertical.

Project the focus perpendicularly to the directrix; the vertex is their midpoint and the axis joins them.

Solution walkthrough

01

Project the focus to the directrix

\[F=(0,2);~y=-2~->~Q=(0,-2)\]

The directrix is horizontal, so its perpendicular direction is vertical. Keeping the focus's x-coordinate and moving to y=-2 gives the perpendicular foot Q.

02

Take the midpoint for the vertex

\[V=((0+0)/2,(2+(-2))/2)=(0,0)\]

A parabola's vertex lies halfway between the focus and directrix along the perpendicular axis because it is equally distant from both.

03

Determine axis, p, and opening

\[\text{axis}:~x=0;~p=2-0=2;~\text{opening}:~\text{up}\]

The axis is the vertical line through F and V. Signed p is the directed vertex-to-focus displacement; it is positive because the focus is above the vertex, so the parabola opens upward toward the focus.

04

Check the defining equality

\[VF=|2-0|=2;~\text{dist}(V,y=-2)=|0-(-2)|=2\]

The vertex is two units from the focus and two perpendicular units from the directrix, confirming the midpoint construction and all reported features.

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Another way

  1. Average the focus y-coordinate 2 and the directrix level -2 to get the vertex level 0, then use the shared x-coordinate 0 for the axis.

!

Common mistake

The horizontal directrix y=-2 is not the axis. The axis must be perpendicular to it, so the axis is the vertical line x=0.

Answer features: focus (0,2) and directrix y=−2 have midpoint vertex (0,0), vertical axis x=0, p=2, and an upward-opening parabola.
The vertex is the midpoint between focus and directrix along the symmetry axis.

Solution walkthrough video