The latus rectum is a special term defined for **the conic section**. To know what a latus rectum is, it helps to know what conic sections are. Conic sections are two-dimensional curves formed by the *intersection of a cone with a plane*. They include parabolas, hyperbolas, and ellipses. Circles are a special case of ellipse.

Index

## Terminology in Conic Section

Here we discuss some important terminology for conic sections so that we can understand the main concept better.

### Focus

The focus ‘or’ foci are* fixed points* from which the conic sections are constructed. Ellipses and hyperbolas have two foci, while circles and parabolas have one focus each.

### Directrix

The directrix is a *fixed line* that is used to construct a conic section. Every point on the conic is at a certain distance \(PF\) from the focus and \(PM\) from the directrix. The ratio of these distances, \(\frac{PF}{PM}\) is called the **eccentricity** of the conic and determines its type.

### Transverse Axis

The axis *passing through the foci and connecting the vertices* of a conic section is said to be the major axis. In the case of the ellipse, this is also called the **major axis**.

### Conjugate Axis

The axis *passing through the centre and perpendicular to the transverse axis* is said to be the conjugate axis. In the case of an ellipse, this axis is called the **minor axis**.

### Latus Rectum

The latus rectum is the **chord** that passes through the **focus** of a conic section, and is *parallel to the directrix*. For each conic section, the latus rectum has a definite length that can be written in terms of axis lengths as given below

- In case of a
**circle,**the length of latus rectum is equal to the**diameter of circle**. - For an
**ellipse**, the length is equal to twice the square of the length of conjugate axis, divided by the length of transverse axis. - In a
**parabola**, the length is simply four times its focal length. - In a
**hyperbola**, the length is equal to twice the square of the length of transverse axis divided by the length of conjugate axis.

## Formulae

The formulae for each of the cases can be seen as follows.

### Ellipse

Let us consider a standard ellipse, *centred on the origin*, and *major axis along the x-axis*. Its equation is given by

\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Then we have,

- Length of major axis = \(2a\)
- Length of minor axis = \(2b\)
- Separation of foci = \(2c\)

Clearly, the foci are located at \((\pm c, 0)\). Then, we have to find where the ellipse is cut by a chord parallel to the y-axis. Let \(y_0\) be the y-coordinate. We have,

\(\begin{align}

\frac {c^2}{a^2} + \frac{{y_0}^2}{b^2} & = 1 \\

\Rightarrow \frac{{y_0}^2}{b^2} & = 1 – \frac {c^2}{a^2} \\

\Rightarrow \frac{{y_0}^2}{b^2} & = \frac {a^2 -c^2}{a^2} \\

\Rightarrow \frac{{y_0}^2}{b^2} & = \frac {b^2}{a^2} \\

\Rightarrow {y_0}^2 & = \frac {b^4}{a^2}\\

\Rightarrow y_0 & = \frac {b^2}{a}

\end{align}

\)

Where we have used the relation \(c = \sqrt{a^2-b^2}\).

As the latus rectum is simply equal to twice of this y-value, we have,

\(\text{Length of latus rectum of ellipse} = 2y_0 = \frac {2b^2}{a}\)

Thus, the **length of **it for** an **ellipse is simply twice the square of the length of the conjugate axis, divided by the length of the transverse axis.

### Circle

As the circle is a special case of the ellipse, with *major and minor axis equal in length of the radius \(r\)*, we have,

\(\text{Length of latus rectum} = \frac {2b^2}{a} = \frac {2r^2}{r} = 2r\)

But as \(2r = d\), the diameter of the circle, we have,

\(\text{Length of latus rectum of circle} = d\)

### Parabola

Let us consider a **standard parabola**, with *a *major axis along *the x-axis.* Its equation is \(y^2 = 4ax\). The focus is at \((a,0)\). So we draw a line through it, parallel to the y-axis. Let it cut the parabola at \((a,y_0)\).

We have,

\(\begin{align}

{y_0}^2 & = 4a \times a \\

& = 4a^2 \\

\Rightarrow y_0 &= 2a

\end{align}

\)

So the length of the latus rectum must be *twice this value*, which means,

\(\text{Length of latus rectum of parabola} = 4a\)

### Hyperbola

The situation here is very similar to the ellipse. The only difference is that the hyperbola’s equation is either \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} – \frac{x^2}{b^2} = 1\). The signs depend on which axis is the transverse axis.

Further, \(c^2 = a^2 + b^2\).

Proceeding with a similar derivation as before, we have the result,

\(\text{Length of latus rectum of hyperbola} = \frac {2b^2}{a}\)

**Also Read:** Hyperbola vs Parabola

## Examples

**Question 1.** Find the length of latus rectum of a parabola, with focus at (0,2) and directrix as \(y = -2\).

**Solution.** This parabola has major axis along y-axis. This means that the focus is at \((0,a)\).

Comparing with given focus, we get, \(a = 2\).

As length of latus rectum is \(4a\), the length in this case is \(4 \times 2 = 8\) units.

**Question 2.** Find the equation of the ellipse with length of latus rectum as 2 units, and focii at \((\pm 2\sqrt{3} , 0)\).

**Solution.** This ellipse has major axis along x-axis.

We know, in such a case,

Length of latus rectum of ellipse = \(\frac {2b^2}{a}\)

Thus, \(\frac {2b^2}{a} = 2\) Or, \(b^2 = a\)

Further, the foci are located at \((\pm c, 0)\). Equating to given values, we have, \(c = 2\sqrt{3}\)

But, we know that \(c^2 = a^2 – b^2\), so,

\({2\sqrt{3}}^2 = a^2 – b^2\)

Using \(b^2 = a\) from above, we have, \(a^2 – a = 12\)

Solving for a, and taking only positive value, we have, \(a = 4\)

Further, \(b = 2\)

Thus the equation of the ellipse must be,

\(\frac{x^2}{16}+\frac{y^2}{4} = 1\)

## FAQs

**What is the formula of latus rectum?**The formula of latus rectum depends on which conic we are talking about. *Using all standard equations, the length of latus rectum: *

1. For ellipse = \(2 \frac{b^2}{a}\)

2. For circle = \(2r\)

3. For parabola = \(4a\)

4. For hyperbola = \(2 \frac{b^2}{a}\)

**Do all conic sections have latus rectum?****Yes,** all conic sections have latus rectum. It is the chord that passes through the focus, and is parallel to the directrix.

**What are the endpoints of the latus rectum?**In standard form, with the major or transverse axis being x-axis, the endpoints *share the same x-coordinate as the focus*. The y-coordinates are found by substituting the x-value in the equation of conic section.

If the y-axis is the major axis, the case is reversed. The* y-coordinate of the endpoints and the focus are the same*. The x-coordinate can be found as above.

**How do you find the equation of a parabola, if length of latus rectum is given?**In standard form, the length of latus rectum is \(4a\), for all parabolas. Now we need to know **two things: **

1. What is the axis of the parabola?

2. What direction does it open in?

Based on that, we have four possible equations:

If x-axis is the major axis, and it opens along +x, the equation is \(y^2 = 4ax\).

And if x-axis is the major axis and it opens along -x, the equation is \(y^2 = -4ax\).

If y-axis is the major axis and it opens along +y, the equation is \(x^2 = 4ay\).

And if y-axis is the major axis and it opens along -y, the equation is \(x^2 = -4ay\).