General Equation of an Ellipse

An ellipse can be defined as the locus of all points that satisfy the equation

where:
x,y are the coordinates of any point on the ellipse,
a, b are the radius on the x and y axes respectively, ( * See radii note below )

This equation is very similar to the one used to define a circle, and much of the discussion is omitted here to avoid duplication. See Basic equation of a circle and General equation of a circle as an introduction to this topic.

The only difference between the circle and the ellipse is that in an ellipse, there are two radius measures, one horizontally along the x-axis, the other vertically along the y-axis. Clearly, for a circle both these have the same value.

By convention, the y radius is usually called b and the x radius is called a.

Ellipses centered at the origin

If the ellipse is centered on the origin, ( its center at 0,0 ) the equation is where
a is the radius along the x-axis ( * See radii note below )
b is the radius along the y-axis

Note that the equations on this page are true only for ellipses that are aligned with the coordinate plane, that is, where the major and minor axes are parallel to the coordinate system.

In the applet above, drag one of the four orange dots around the ellipse to resize it, and note how the equation changes to match.

Ellipses not centered at the origin

Just as with the circle equations, we subtract offsets from the x and y terms to translate (or "move") the ellipse back to the origin. So the full form of the equation is where
a is the radius along the x-axis
b is the radius along the y-axis
h, k are the x,y coordinates of the ellipse's center.

In the applet above, drag the orange dot at the center to move the ellipse, and note how the equations change to match.

Derivation

Start with the basic equation of a circle: Divide both sides by r²: Replace the radius with the a separate radius for the x and y axes:

A circle is just a particular ellipse

In the applet above, click 'reset' and drag the right orange dot left until the two radii are the same. This is a circle, and you will get an equation that looks like where r is whatever radius you set up. If you multiply through by r² you get which is the general equation of a circle. This demonstrates that a circle is just a special case of an ellipse.

Other forms of the equation

Using trigonometry to find the points on the ellipse, we get another form of the equation. For more see Parametric equation of an ellipse

Things to try

In the above applet click 'reset', and 'hide details'.
Drag the five orange dots to create a new ellipse at a new center point.
Write the equations of the ellipse in general form.
Click "show details" to check your answers.

* Note on radii

In many textbooks, the two radii are specified as being the semi-major and semi-minor axes. Recall that these are the longest and shortest radii of the ellipse respectively. The trouble with this is that if the ellipse is tall and narrow, they have to be reversed, so you wind up with two forms of the equations, one for tall thin ellipses and another for short wide ones.

Regardless of what you call these radii, remember that the x term must use the radius along the x-axis, and the y term must use the radius along the y-axis: