What is an Ellipse Equation?

An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the distances to the two focal points is constant. In coordinate geometry, the standard equation of an ellipse helps us identify its center, orientation, and how wide or tall it is relative to its axes.

Standard Form Equations

Depending on the orientation, the equation takes one of two forms:

• Horizontal Major Axis: ((x-h)² / a²) + ((y-k)² / b²) = 1
• Vertical Major Axis: ((x-h)² / b²) + ((y-k)² / a²) = 1

Where (h, k) is the center, 'a' is the semi-major axis, and 'b' is the semi-minor axis. Note that by definition, a > b.

How to Use This Ellipse Calculator

Using our Ellipse Equation Calculator with steps is straightforward. Simply input the center coordinates (h and k) and the lengths of the semi-axes (a and b). Select whether the ellipse is oriented horizontally or vertically. Our tool will instantly generate the final equation and provide the coordinates for the vertices, co-vertices, and foci, along with the focal distance (c) calculation.

Frequently Asked Questions (FAQ)

What is the difference between 'a' and 'b' in an ellipse?

'a' represents the semi-major axis, which is the longest radius of the ellipse. 'b' represents the semi-minor axis, the shortest radius. In standard geometry problems, 'a' is always greater than or equal to 'b'.

How do you find the foci of an ellipse?

The distance from the center to the foci is denoted as 'c'. It is calculated using the formula c² = a² - b². Once 'c' is found, you add and subtract it from the center coordinates along the major axis.

Can the center be negative?

Yes, the center (h, k) can be any real coordinate on the Cartesian plane. The calculator handles positive, negative, and zero values for the center points.