The equation of a vertical ellipse is x^2b^2+ y^2a^2=1. The equation of a horizontal ellipse is x^2a^2+ y^2b^2=1. In both cases, a and b are positive numbers with a>b.
x^2/18+y^2/20=1
Practice makes perfect
We want to write an equation of an ellipse in standard form with center at the origin. Before we consider the given characteristics of the ellipse, let's think about how a and b relate to the width and height of an ellipse centered at the origin.
Vertices:& (0,± a) or (± a, 0)
Co-vertices:& (0,± b) or (± b, 0)
Because the vertices and co-vertices extend from the negative to the positive values of a and b, the length of the major and minor axes will always be 2a and 2b. Let's look at an example!
We are told that a= 2sqrt(5) and b= 3sqrt(2). Let's use this information to calculate the length of the major and minor axes.
Axes of the Ellipse
Major Axis
Minor Axis
2 a
2 b
2( 2sqrt(5))
2( 3sqrt(2))
4sqrt(5)
6sqrt(2)
We are also told that the width of the ellipse is 6sqrt(2). This matches the length of the minor axis. Therefore, the length of the major axis is the height of the ellipse. Since the major axis is the height and the minor axis the width, we have a vertical ellipse.
x^2/b^2+y^2/a^2=1, a>b>0
Let's substitute the given values into the above equation and simplify.
