# Latus Rectum: Parabola, Ellipse, and Hyperbola Formulas

TL;DR

The latus rectum is the focal chord of a conic drawn perpendicular to its main axis, ending on the curve; its length is 4a for a parabola \(y^2=4ax\) and \(\frac{2b^2}{a}\) for both an ellipse and a hyperbola. This article defines the latus rectum, gives each conic's formula and endpoint coordinates, explains why it measures a conic's width at the focus, and works through examples.

## The One Measurement That Tells You How Wide A Curve Opens At Its Focus

The **latus rectum** of a conic section is the **chord that passes through a focus, runs perpendicular to the major (or transverse) axis, and has both endpoints on the curve**. It gives a direct measure of how wide the conic is at the focus. A parabola has one latus rectum; an ellipse and a hyperbola each have two, one through each focus. The latus rectum is a defining feature of the [conic sections](/content/math/geometry/conic-sections/index.html) family, and it is tied closely to a conic's [eccentricity](/content/math/geometry/eccentricity/index.html).

## Latus Rectum Of A Parabola: Length 4a

For the standard parabola opening rightward:

\[y^2=4ax\]

the focus is at \((a,0)\) and the directrix is the line \(x=-a\). The latus rectum is the vertical chord through the focus. To find where it meets the curve, set \(x=a\):

\[y^2=4a(a)=4a^2\Rightarrow y=\pm 2a\]

So the endpoints are \(L(a,2a)\) and \(L'(a,-2a)\), and the length is the distance between them:

\[\text{Latus rectum} = 2a - (-2a) = 4a\]

Here \(a\) is the distance from the vertex to the focus. A larger \(a\) opens the parabola wider, and the latus rectum \(4a\) measures exactly that opening at the focus. The focus itself is the [focus of a parabola](/content/math/geometry/focus-of-parabola/index.html), and the guiding line is the [directrix of a parabola](/content/math/geometry/directrix-of-parabola/index.html).

## Latus Rectum Of An Ellipse: Length \(\frac{2b^2}{a}\)

For the standard horizontal ellipse:

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b\]

the foci sit at \((\pm ae,0)\), where \(e\) is the eccentricity and \(0<e<1\). Substitute the focal x-value \(x=ae\) into the equation and solve for y; the algebra collapses to \(y=\pm \frac{b^2}{a}\). So the endpoints of the latus rectum through the focus \((ae,0)\) are:

\[\left(ae, \frac{b^2}{a}\right) \quad\text{and}\quad \left(ae, -\frac{b^2}{a}\right)\]

and the length is:

\[\text{Latus rectum} = \frac{2b^2}{a}\]

Here \(a\) is the semi-major axis and \(b\) is the semi-minor axis. Because an ellipse has two foci, it has two latus rectums, each of the same length \(\frac{2b^2}{a}\). These pass through the two [foci of the ellipse](/content/math/geometry/foci-of-ellipse/index.html).

## Latus Rectum Of A Hyperbola: Length \(\frac{2b^2}{a}\)

For the standard horizontal hyperbola:

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

the foci sit at \((\pm ae,0)\), where the eccentricity \(e>1\). Substituting \(x=ae\) and solving for y again gives \(y=\pm \frac{b^2}{a}\), so the latus-rectum endpoints through \((ae,0)\) are:

\[\left(ae, \frac{b^2}{a}\right) \quad\text{and}\quad \left(ae, -\frac{b^2}{a}\right)\]

and the length is the same expression as the ellipse:

\[\text{Latus rectum} = \frac{2b^2}{a}\]

The two conics share the formula because both use \(b^2\) tied to the focal geometry; the difference lives in the equation's sign and in the range of \(e\), not in the latus-rectum length. The two chords pass through the two [foci of the hyperbola](/content/math/geometry/foci-of-hyperbola/index.html).

## Examples of Latus Rectum

### Example 1

**Find the length and endpoints of the latus rectum of the parabola \(y^2=12x\).**

Compare with \(y^2=4ax\):

\[4a=12\Rightarrow a=3\]

Length of latus rectum:

\[4a=12\]

Endpoints, using \((a,\pm 2a)\):

\[(3,6)\text{ and } (3,-6)\]

The latus rectum has length 12, with endpoints \((3,6)\) and \((3,-6)\).

### Example 2

**A student reports the latus rectum of \(y^2=12x\) as \(a=3\). Spot the error.**

A natural first move is to read off \(a=3\) and stop, treating \(a\) as the answer. But \(a\) is only the focus-to-vertex distance, not the chord length, and calling it the latus rectum confuses a coordinate with a length.

The latus-rectum length is 12, so:

\[4a=12 \Rightarrow a=3\]

### Example 3

**Find the length of the latus rectum of the ellipse \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).**

Read off \(a^2=25\) and \(b^2=9\), so \(a=5\) and \(b=3\). Since \(a>b\), the major axis is horizontal, and the formula applies directly:

\[\text{Latus rectum}=\frac{2b^2}{a}=\frac{2(9)}{5}=\frac{18}{5}=3.6\]

### Example 4

**Find the length of the latus rectum of the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\).**

Read off \(a^2=16\) and \(b^2=9\), so \(a=4\) and \(b=3\):

\[\text{Latus rectum}=\frac{2b^2}{a}=\frac{2(9)}{4}=\frac{18}{4}=4.5\]

### Example 5

**An ellipse has latus rectum \(\frac{2b^2}{a} = 8\) and semi-major axis \(a=4\). Find \(b\).**

Substitute the known values:

\[\frac{2b^2}{4} = 8\Rightarrow b^2=16 \Rightarrow b=4\]

### Example 6

**A parabolic satellite dish is modelled by \(y^2=4ax\) and must have a latus rectum of 2 metres. Find \(a\) and the focus position.**

The latus rectum is \(4a\):

\[4a=2\Rightarrow a=0.5 \text{ m}\]

The focus sits at \((a,0)=(0.5,0)\) which is 0.5 m from the vertex along the axis.

## Where The Latus Rectum Earns Its Keep: Width At The Focus

The latus rectum matters because it converts an abstract focus into a concrete size: how wide the curve is right where the action happens.

- **Optics and antennas.** A parabolic mirror or dish focuses signal at its focus; the latus rectum fixes how broad the beam is there.

- **Orbits.** In planetary motion, the semi-latus rectum \(\frac{b^2}{a}\) is the standard parameter for an orbit's shape.

- **Why it is the natural width.** The destination is a _focus-anchored size_.

## Key Takeaways

- The **latus rectum** is the focal chord perpendicular to the main axis, ending on the curve.

- For a parabola \(y^2=4ax\), its length is \(4a\), with endpoints \((a,\pm 2a)\).

- For an ellipse and a hyperbola, its length is \(\dfrac{2b^2}{a}\), with endpoints \((ae,\pm \dfrac{b^2}{a})\).

- A parabola has one latus rectum; the ellipse and hyperbola each have two.

- The latus rectum measures a conic's width at the focus.