Question

Find an equation of the ellipse that has vertices (0,±5) and foci (0,±3).

Short answer

The equation of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

Step-by-step solution

Identify the Major and Minor Axes

First, observe that the vertices and foci are on the y-axis, indicating a vertically oriented ellipse. The general form of the ellipse equation is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) where \(a > b\).

Determine the Length of the Semi-major Axis (a)

The distance from the center to a vertex is the semi-major axis (a). Given the vertices are (0, ±5), \(a = 5\).

Determine the Distance to the Foci (c)

The distance from the center to a focus is given as the foci (0, ±3), so \(c = 3\).

Use the Relationship Between a, b, and c

In ellipses, the relationship between the semi-major axis (a), semi-minor axis (b), and the focal distance (c) is given by \(a^2 = b^2 + c^2\). Substituting the known values: \(25 = b^2 + 9\).

Solve for the Semi-minor Axis (b)

Rearrange the equation and solve for \(b^2\): \(b^2 = 16\) which implies that \(b = 4\).

Write the Equation of the Ellipse

Substitute \(a\) and \(b\) into the general form of the ellipse equation: \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

Key concepts

Semi-Major Axis

The semi-major axis is one of the fundamental parts of an ellipse. It is the longest radius of the ellipse and extends from its center to a vertex.

In this exercise, we are given the vertices \(0, \pm5\), which means the distance from the center to each of these points is 5 units. So, we have \(a = 5\).

In general, the semi-major axis (\