How to Write an Equation of an Ellipse in Standard Form with Given Vertices?

To find the equation of an ellipse in standard form with the center at the origin and given vertices, we first need to identify some essential characteristics based on the provided information. Here, the vertex is at (30, 0) and the co-vertex is at (0, 2).

The standard form of an ellipse centered at the origin is:

(x² / a²) + (y² / b²) = 1

or

(y² / a²) + (x² / b²) = 1

Where:

• a is the distance from the center to a vertex,
• b is the distance from the center to a co-vertex.

Given the vertex at (30, 0), the distance a = 30. Thus, a = 30, which means that a² = 900.

Given the co-vertex at (0, 2), the distance b = 2. Thus, b = 2, which means that b² = 4.

Since the vertices are aligned along the x-axis (horizontal), we will use the first standard form equation:

(x² / 900) + (y² / 4) = 1

So the equation of the ellipse in standard form is:

(x² / 900) + (y² / 4) = 1

This equation represents an ellipse situated horizontally with a center at the origin.