Vertex of an Ellipse: Definition, Formula & Examples

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.

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