# How do I find the major and minor axes of an ellipse?

Mar 7, 2015

Assuming that the axes have not been rotated:

In the standard form equation, look at the numbers in the denominators. They are the squares of half the lengths of the axes of the ellipse parallel to the respective variable.
If the number under the fraction involving ${\left(x - h\right)}^{2}$ is larger than the number under the other fraction, then the major axis of the ellipse is parallel to the $x$-axis of the coordinate system. And vice versa.

From standard form for the equation of an ellipse:
${\left(x - h\right)}^{2} / \left({a}^{2}\right) + {\left(y - k\right)}^{2} / \left({b}^{2}\right) = 1$

The center of the ellipse is $\left(h , k\right)$

The major axis of the ellipse has length = the larger of $2 a$ or $2 b$ and the minor axis has length = the smaller.

If $a > b$ then the major axis of the ellipse is parallel to the $x$-axis (and, the minor axis is parallel to the $y$-axis)
In this case the endpoints of the major axis are $\left(h - a , k\right)$ and $\left(h + a , k\right)$ and the endpoints of the minor axis are $\left(h , k - b\right)$ and $\left(h , k + b\right)$

if $a < b$ then the major and minor axes of the ellipse with respect to the $x$ and $y$-axes are reversed (the dual)
if $a < b$ the major axis is parallel to the $y$-axis (and the minor axis is parallel to the $x$-axis)
In this case the endpoint of the minor axis are $\left(h - a , k\right)$ and $\left(h + a , k\right)$ and the endpoints of the major axis are $\left(h , k - b\right)$ and $\left(h , k + b\right)$

By the way: if $a = b$, then the "ellipse" is a circle.

Example:
${\left(x - 3\right)}^{2} / \left(4\right) + {\left(y + 2\right)}^{2} / \left(49\right) = 1$

Major axis: parallel to $y$-axis
Lengths: major axis length is $7$, minor has length $2$

Center: $\left(3 , - 2\right)$

Endpoints of axes:
(parallel to the $x$-axis) : $\left(1 , - 2\right)$ and $\left(5 , - 2\right)$ -- minor
(parallel to the $y$-axis) : $\left(10 , 3\right)$ and $\left(- 4 , 3\right)$ -- major