Ximpledu
English

Parabola

See how to solve a parabola
(focus, directrix, equation, latus rectum).
8 examples and their solutions.

Definition

A parabola is the set of points that satisfy
(point ~ focus) = (point ~ directrix).
Focus: blue point
Directrix: dashed line

Parabola: y2 = 4px

Equation

y2 = 4px



Focus: (p, 0)
Directrix: x = -p

Example

y2 = 8x
1. Focus?
2. Equation of the directrix?
Solution

Example

Focus: (3, 0)
Directrix: x = -3
Equation of the parabola?
Solution

Example

Focus: (3, 4)
Directrix: x = -1
Equation of the parabola?
Solution

Latus Rectum

y2 = 4px



(latus rectum) = |4p|
The latus rectum of a parabola is a segment
that passes through the focus
and that is parallel to the directrix.

Example

y2 = 8x
Latus rectum?
Solution

Example

y2 = -12x
Latus rectum?
Solution

Parabola: x2 = 4py

Equation

x2 = 4py



Focus: (0, p)
Directrix: y = -p

Example

y = x2
1. Focus?
2. Equation of the directrix?
Solution

Example

Focus: (0, 2)
Directrix: y = -2
Equation of the parabola?
Solution

Latus Rectum

x2 = 4py



(latus rectum) = |4p|

Example

y = 3x2
Latus rectum?
Solution