Equations of Ellipse and Hyperbola An ellipse is a plane curve with two focused points, which we call the vertices . The distance between the vertices...
An ellipse is a plane curve with two focused points, which we call the vertices. The distance between the vertices is equal to the length of the major axis, and the distance from the center to either vertex is equal to the length of the minor axis.
An hyperbola has two branches that approach the central axis but are not part of the curve itself. The center is located between the two branches. The distance between the center and either vertex is equal to the length of the major axis, while the distance between the center and the foci is equal to the length of the minor axis.
Equations:
An equation for an ellipse is:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where:
(h, k) is the center point
a is the length of the major axis
b is the length of the minor axis
An equation for a hyperbola is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where:
(h, k) is the center point
a is the length of the major axis
b is the length of the minor axis
Examples:
Ellipse: (x - 2)^2/9 + (y - 3)^2/4 = 1
Hyperbola: (x + 3)^2/25 - (y - 1)^2/9 = 1
By understanding the equations of these two types of curves, we can plot them on coordinate graphs and analyze their properties