Parabola: Latus Rectum

Parabola: Latus Rectum

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

Formula 1

If the parabola is y^2 = 4px, then the length of the latus rectum is |4p|.

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 [y2 = 4px],
then the length of the latus rectum is |4p|.

The latus rectum is the coefficient of the x term.

Parabola - Formula y2 = 4px

Example 1: Latus Rectum of y2 = 8x

Find the latus rectum of the given parabola. y^2 = 8x

The coefficient of the x term is 8.

So (latus rectum) = |8|.

|8| = 8

So (latus rectum) = 8.

This is the graph of y2 = 8x
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?

Find the latus rectum of the parabola whose focus and directrix are below. Focus: (-1, 0), Directrix: x = 1

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
y2 = 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 y2 = -4x
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 x2 = 4py

If the parabola is x^2 = 4py, then the length of the latus rectum is also |4p|.

If the parabola is [x2 = 4py],
then the length of the latus rectum is also |4p|.

The latus rectum is the coefficient of the y term.

Parabola - Formula x2 = 4py

Example 3: Latus Rectum of x2 = 12y

Find the latus rectum of the given parabola. x^2 = 12y

The coefficient of the y term is 12.

So (latus rectum) = |12|.

|12| = 12

So (latus rectum) = 12.

This is the graph of x2 = 12y
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 = -x2

Find the latus rectum of the given parabola. y = -x^2

Switch both sides:
-x2 = y.

Multiply -1 on both sides.
Then x2 = -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 = -x2
and its latus rectum.

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

And the length of the latus rectum is 1.