# Parabola: Latus Rectum

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

## Formula 1

The latus rectum of a parabola is a segment

that passes through the focus

and that is perpendicular to the axis of symmetry.

Parabola: formula, focus, directrix

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

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

|4*p*| is the absolute value of the *x*'s coefficient.

## Proof: Formula 1

The focus is on the latus rectum.

So put *p* into the parabola's *x*.

Then *y* = ±|2*p*|.

So the endpoints of the latus rectum are

(*p*, +|2*p*|) and (*p*, -|2*p*|).

So the latus rectum is |2*p*| + |2*p*| = |4*p*|.

## Example 1

(latus rectum) = |4*p*|

= |4⋅2|

= 8

The red segment is the latus rectum,

whose length is |4⋅2| = 8.

## Formula 2

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

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

|4*p*| is the absolute value of the *y*'s coefficient.

## Proof: Formula 2

The focus is on the latus rectum.

So put *p* into the parabola's *y*.

Then *x* = ±|2*p*|.

So the endpoints of the latus rectum are

(+|2*p*|, *p*) and (-|2*p*|, *p*).

So the latus rectum is |2*p*| + |2*p*| = |4*p*|.

## Example 2

(latus rectum) = |4*p*|

= |4⋅(-3)|

= 12

The red segment is the latus rectum,

whose length is |4⋅(-3)| = 12.