Mathematics.

conic sections

Parabola

Analytic Geometry20 minDifficulty4 out of 10

You should know: conic sections

Overview

A parabola is a plane curve that is mirror-symmetric and approximately U-shaped. It arises as a conic section with eccentricity exactly 1, and equivalently as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Parabolas are the graphs of quadratic functions and appear throughout physics as the shape of projectile trajectories.

Formal Definition

Definition

A parabola with vertex at the origin, opening along the x-axis (or, in the second form, along the y-axis), in standard position:

y2=4pxy^2 = 4px

Focus at (p, 0); directrix x = -p

Standard form, opens along x-axis
x2=4pyx^2 = 4py

Focus at (0, p); directrix y = -p

Standard form, opens along y-axis
∣PF∣=d(P,directrix)|PF| = d(P, \text{directrix})

Focus-directrix definition: eccentricity e = 1

Properties

Vertex

The point (0,0) in standard position, midway between focus and directrix\text{The point } (0,0) \text{ in standard position, midway between focus and directrix}

Focus

(p,0) for y2=4px, or (0,p) for x2=4py(p, 0) \text{ for } y^2=4px \text{, or } (0,p) \text{ for } x^2 = 4py

Directrix

x=−p (or y=−p), a fixed line the same distance from the vertex as the focus, on the opposite sidex = -p \text{ (or } y=-p\text{)}, \text{ a fixed line the same distance from the vertex as the focus, on the opposite side}

Eccentricity

e=1 always, for every parabolae = 1 \text{ always, for every parabola}

Axis of symmetry

The line through the vertex and focus, perpendicular to the directrix\text{The line through the vertex and focus, perpendicular to the directrix}

Worked Examples

  1. Match to standard form y² = 4px, so 4p = 12, giving p = 3.

    4p=12  ⟹  p=34p = 12 \implies p = 3

  2. The focus is at (p, 0) and the directrix is x = -p.

    F=(3,0),x=−3F = (3, 0), \quad x = -3

Answer: Focus (3, 0); directrix x = -3.

Practice Problems

Difficulty 4/10

Find the focus and directrix of x² = -8y.

Common Mistakes

Common Mistake

Mixing up which variable is squared and assuming the parabola always opens upward.

y² = 4px opens left/right (along the x-axis) while x² = 4py opens up/down (along the y-axis) — the squared variable tells you the axis of symmetry, and the sign of p tells you the direction.

Summary

  • A parabola is the set of points equidistant from a focus and a directrix — the unique conic with eccentricity e = 1.
  • Standard forms: y² = 4px (opens along x-axis, focus (p,0), directrix x=-p) or x² = 4py (opens along y-axis, focus (0,p), directrix y=-p).
  • The vertex sits midway between focus and directrix, on the axis of symmetry.
  • Parabolas model projectile motion and are the shape of graphs of quadratic functions.

References

  1. WebsiteWikipedia — Parabola