In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

Verified step by step guidance

1. Identify the center of the ellipse. Since the foci are at (-2, 0) and (2, 0), the center is at the midpoint of these points, which is (0, 0).
2. Determine the distance between the center and each focus, which is the value of c. Here, c = 2.
3. Use the y-intercepts to find the value of b. Since the y-intercepts are -3 and 3, the distance from the center to the intercepts is 3, so b = 3.
4. Use the relationship c^2 = a^2 - b^2 to find a^2. Substitute c = 2 and b = 3 into the equation to solve for a^2.
5. Write the standard form of the equation of the ellipse using the values of a^2 and b^2. The standard form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Key Concepts

Ellipse Definition

An ellipse is a set of points in a plane where the sum of the distances from two fixed points, called foci, is constant. The standard form of an ellipse's equation can be expressed as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, a is the distance from the center to the vertices along the x-axis, and b is the distance along the y-axis.

Foci and Vertices

The foci of an ellipse are two fixed points that help define its shape. For an ellipse centered at the origin with horizontal major axis, the foci are located at (±c, 0), where c² = a² - b². The vertices are the endpoints of the major axis, located at (±a, 0) for horizontal ellipses, and the minor axis endpoints are at (0, ±b).

Y-Intercepts

Y-intercepts are points where the graph intersects the y-axis, which occur when x = 0. For the given ellipse, the y-intercepts at -3 and 3 indicate that the ellipse extends vertically to these points. This information helps determine the value of b in the standard form equation, as it represents the distance from the center to the vertices along the y-axis.