# Vertex of an Ellipse: Definition, Formula & Examples

TL;DR

The vertices of an ellipse are the two endpoints of the major axis - the farthest points from the center - located a distance **a** from the center along the longer axis. The endpoints of the shorter (minor) axis are the co-vertices, at distance **b**. This article defines both, gives the formulas for finding them from an ellipse's equation, and works through examples.

## What Is The Vertex Of An Ellipse?

An **ellipse** is the smooth oval curve of points whose distances from two fixed points (the **foci**) add to a constant. The **vertices** of an ellipse are the **two endpoints of the major axis** \- the longest diameter of the ellipse - and they are the two points on the curve farthest from the center. Each endpoint of the shorter **minor axis** is called a **co-vertex**. An ellipse therefore has exactly **two vertices and two co-vertices**.

The key distinction, and the source of nearly every mistake on this topic, is that **vertices lie on the major (longer) axis** while **co-vertices lie on the minor (shorter) axis**. Since the major axis is longer, the vertices are always farther from the center than the co-vertices - that is, **a > b** always.

## The Formula: Finding The Vertices

For an ellipse **centered at the origin**, the standard equation is:

**Horizontal ellipse** (major axis along the x-axis):

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \Rightarrow \quad \text{vertices } (\pm a, 0), 	ext{ co-vertices } (0, \pm b) \]

**Vertical ellipse** (major axis along the y-axis):

\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad \Rightarrow \quad \text{vertices } (0, \pm a), 	ext{ co-vertices } (\pm b, 0) \]

For an ellipse **centered at (h,k)**, shift everything by the center:

\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad \Rightarrow \quad \text{vertices } (h \pm a, k) \]

The recipe is always the same three steps: put the equation in standard form, find the **larger** denominator and call its square root **a**, then place the vertices **a** units from the center **along the axis of the larger denominator**. The distance **a** is the **semi-major axis**; **b** is the **semi-minor axis**.

## Examples Of The Vertex Of An Ellipse

### Example 1

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

The larger denominator is 25, so the ellipse is horizontal and **a = 5**.

**Vertices:** (±5,0)

**Co-vertices:** (0,3) and (0,-3).

### Example 2

**Find the vertices of** \( \frac{x^2}{16} + \frac{y^2}{49} = 1. **

The larger denominator is 49, so the major axis is **vertical** and **a = 7**.

**Vertices:** (0,±7)

**Co-vertices:** (±4,0).

### Example 3

**Find the vertices of** \( \frac{x^2}{9} + \frac{y^2}{36} = 1. **

The larger denominator is 36, so the major axis is vertical, giving **a = 6**.

**Vertices:** (0,±6)

**Co-vertices:** (±3,0).

### Example 4

**Center at (2,-1)** with equation \( \frac{(x-2)^2}{49} + \frac{(y+1)^2}{4} = 1. **

The larger denominator is 49, so the major axis is horizontal, giving **a = 7**.

**Vertices:** (2±7,-1) = (9,-1) and (-5,-1).

### Example 5

**The vertices of an ellipse are (0,8) and (0,-8)**, and the co-vertices are (5,0) and (-5,0).

**Standard equation:** \( \frac{x^2}{25} + \frac{y^2}{64} = 1 **.

### Example 6

**A whispering-gallery dome has an elliptical cross-section 40 m wide and 24 m tall.**

**Vertices:** (±20,0) = (20,0) and (-20,0).

## Where The Vertex Of An Ellipse Earns Its Keep

The vertices fix the size and orientation of every real ellipse, so they matter wherever ellipses appear.

- **Planetary orbits:** Planets travel in ellipses with the Sun at one focus.
- **Whispering galleries:** In an elliptical dome, sound reflects from one focus to another.
- **Engineering and optics:** Elliptical gears, arches, and reflectors depend on their axes.

## The Mistakes Students Make Most Often

### Mistake 1

**Assuming the x-term always gives the vertices**

**Correct way:** Compare the two denominators first; the **larger** one marks the major axis; the vertices lie on _that_ axis.

### Mistake 2

**Swapping vertices and co-vertices.**

**Correct way:** Vertices are on the major axis, co-vertices are on the minor axis.

### Mistake 3

**Forgetting to shift by the center.**

**Correct way:** For a center at (h,k), add the center coordinates to find vertices correctly.

## Conclusion

- The **vertices of an ellipse** are the two endpoints of the major axis, at distance **a** from the center.
- The co-vertices are at distance **b**, with **a > b** always.
- Find the vertices by identifying the larger denominator and placing **a** units along that axis.