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# 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.

y2 = 8x
Latus rectum?
Solution

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|

y = 3x2
Latus rectum?
Solution