# Parabola

See how to solve a parabola

(focus, directrix, equation, latus rectum).

8 examples and their solutions.

## Definition

(point ~ focus) = (point ~ directrix).

Focus: blue point

Directrix: dashed line

## Parabola: y^{2} = 4px

### Equation

y

Focus: (p, 0)

Directrix: x = -p

^{2}= 4pxFocus: (p, 0)

Directrix: x = -p

### Example

y

1. Focus?

2. Equation of the directrix?

Solution ^{2}= 8x1. Focus?

2. Equation of the directrix?

y

= 4⋅2⋅x

1. (2, 0)

2. x = -2

^{2}= 8x= 4⋅2⋅x

1. (2, 0)

2. x = -2

Close

### Example

Focus: (3, 0)

Directrix: x = -3

Equation of the parabola?

Solution Directrix: x = -3

Equation of the parabola?

Focus: (3, 0)

Directrix: x = -3

y

y

Directrix: x = -3

y

^{2}= 4⋅3⋅xy

^{2}= 12xClose

### Example

Focus: (3, 4)

Directrix: x = -1

Equation of the parabola?

Solution Directrix: x = -1

Equation of the parabola?

2p = 3 + (-1)

2p = 4

p = 2

(2, 0) → (3, 4) = (2 + 1, 0 + 4)

(x, y) → (x + 1, y + 4) - [2]

(y - 4)

^{2}= 4⋅2⋅(x - 1) - [3]

(y - 4)

^{2}= 8(x - 1)

[1]

Draw the parabola, the focus, and the directrix.

The vertex of the parabola

is the midpoint of the focus and the directrix.

So set

(focus ~ vertex) = (vertex ~ directrix) = p.

The vertex of the parabola

is the midpoint of the focus and the directrix.

So set

(focus ~ vertex) = (vertex ~ directrix) = p.

[2]

p = 2

So the focus should be (2, 0).

But the focus is (3, 4).

Then there's a translation

(2, 0) → (3, 4).

(3, 4) = (2 + 1, 0 + 4)

So the translation is

(x, y) → (x + 1, y + 4).

So the focus should be (2, 0).

But the focus is (3, 4).

Then there's a translation

(2, 0) → (3, 4).

(3, 4) = (2 + 1, 0 + 4)

So the translation is

(x, y) → (x + 1, y + 4).

[3]

p = 2

(x, y) → (x + 1, y + 4)

Then the equation of the parabola is

(y - 4)

(x, y) → (x + 1, y + 4)

Then the equation of the parabola is

(y - 4)

^{2}= 4⋅2⋅(x - 1).Close

### Latus Rectum

y

(latus rectum) = |4p|

The latus rectum of a parabola is a segment^{2}= 4px(latus rectum) = |4p|

that passes through the focus

and that is parallel to the directrix.

### Example

y

Latus rectum?

Solution ^{2}= 8xLatus rectum?

(latus rectum) = |8|

= 8

= 8

Close

### Example

y

Latus rectum?

Solution ^{2}= -12xLatus rectum?

(latus rectum) = |-12|

= 12

= 12

Close

## Parabola: x^{2} = 4py

### Equation

x

Focus: (0, p)

Directrix: y = -p

^{2}= 4pyFocus: (0, p)

Directrix: y = -p

### Example

y = x

1. Focus?

2. Equation of the directrix?

Solution ^{2}1. Focus?

2. Equation of the directrix?

y = x

x

= 4⋅14⋅y

1. (0, 14)

2. y = -14

^{2}x

^{2}= 4= 4⋅14⋅y

1. (0, 14)

2. y = -14

Close

### Example

Focus: (0, 2)

Directrix: y = -2

Equation of the parabola?

Solution Directrix: y = -2

Equation of the parabola?

Focus: (0, 2)

Directrix: y = -2

x

x

Directrix: y = -2

x

^{2}= 4⋅2⋅yx

^{2}= 8yClose

### Latus Rectum

x

(latus rectum) = |4p|

^{2}= 4py(latus rectum) = |4p|

### Example

y = 3x

Latus rectum?

Solution ^{2}Latus rectum?

y = 3x

3x

x

(latus rectum) = |13|

= 13

^{2}3x

^{2}= yx

^{2}= 13y(latus rectum) = |13|

= 13

Close