# Parabola: Latus Rectum

How to find the latus rectum of a parabola: formulas, examples, and their solutions.

## Formula 1

The latus rectum of a parabola is a segment

that passes through the focus

and that is parallel to the directrix.

Parabola - Focus

Parabola - Directrix

If the parabola is [*y*^{2} = 4*px*],

then the length of the latus rectum is |4*p*|.

The latus rectum is the coefficient of the *x* term.

Parabola - Formula *y*^{2} = 4*px*

## Example 1: Latus Rectum of *y*^{2} = 8*x*

The coefficient of the *x* term is 8.

So (latus rectum) = |8|.

|8| = 8

So (latus rectum) = 8.

This is the graph of *y*^{2} = 8*x*

and its latus rectum.

The latus rectum passes through the focus (2, 0).

And the length of the latus rectum is 8.

## Example 2: Focus (-1, 0), Directrix *x* = 1, Latus Rectum?

The focus is (-1, 0).

And the directrix is *x* = -(-1).

So *p* = -1.

*p* = -1

And the directrix is [*x* = ...].

Then the equation of the parabola is*y*^{2} = 4⋅(-1)⋅*x*.

Parabola - Formula

4⋅(-1) = -4

The coefficient of the *x* term is -4.

So (latus rectum) = |-4|.

|-4| = 4

So (latus rectum) = 4.

This is the graph of *y*^{2} = -4*x*

and its latus rectum.

The latus rectum passes through the focus (-1, 0).

And the length of the latus rectum is 4.

## Formula: Latus Rectum of *x*^{2} = 4*py*

If the parabola is [*x*^{2} = 4*py*],

then the length of the latus rectum is also |4*p*|.

The latus rectum is the coefficient of the *y* term.

Parabola - Formula *x*^{2} = 4*py*

## Example 3: Latus Rectum of *x*^{2} = 12*y*

The coefficient of the *y* term is 12.

So (latus rectum) = |12|.

|12| = 12

So (latus rectum) = 12.

This is the graph of *x*^{2} = 12*y*

and its latus rectum.

The latus rectum passes through the focus (0, 3).

And the length of the latus rectum is 12.

## Example 4: Latus Rectum of *y* = -*x*^{2}

Switch both sides:

-*x*^{2} = *y*.

Multiply -1 on both sides.

Then *x*^{2} = -1⋅*y*.

The coefficient of the *y* term is -1.

So (latus rectum) = |-1|.

|-1| = 1

So (latus rectum) = 1.

This is the graph of *y* = -*x*^{2}

and its latus rectum.

The latus rectum passes through the focus (0, -1/4).

And the length of the latus rectum is 1.