Parabola: Latus Rectum

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. 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 perpendicular to the axis of symmetry.

Parabola: formula, focus, directrix

If the parabola is y2 = 4px,
then the length of the latus rectum is |4p|.

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

Proof: Formula 1

Parabola - Latus Rectum: Proof of the Formula 1

The focus is on the latus rectum.
So put p into the parabola's x.

Then y = ±|2p|.
So the endpoints of the latus rectum are
(p, +|2p|) and (p, -|2p|).

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

Example 1

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

(latus rectum) = |4p|
= |4⋅2|
= 8

The red segment is the latus rectum,
whose length is |4⋅2| = 8.

Formula 2

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

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

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

Proof: Formula 2

Parabola - Latus Rectum: Proof of the Formula 2

The focus is on the latus rectum.
So put p into the parabola's y.

Then x = ±|2p|.
So the endpoints of the latus rectum are
(+|2p|, p) and (-|2p|, p).

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

Example 2

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

(latus rectum) = |4p|
= |4⋅(-3)|
= 12

The red segment is the latus rectum,
whose length is |4⋅(-3)| = 12.