Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-5, 0), (5, 0); vertices: (-8, 0), (8,0)

Verified step by step guidance

1. Identify the center of the ellipse by finding the midpoint of the vertices. Since the vertices are (-8, 0) and (8, 0), the center is at \(\left( \frac{-8 + 8}{2}, \frac{0 + 0}{2} \right) = (0, 0)\).
2. Determine the orientation of the ellipse. Because the vertices and foci lie on the x-axis, the major axis is horizontal.
3. Calculate the distance from the center to a vertex, which is the value of \(a\). Here, \(a = 8\).
4. Calculate the distance from the center to a focus, which is the value of \(c\). Here, \(c = 5\).
5. Use the relationship \(c^2 = a^2 - b^2\) to find \(b^2\). Rearranged, \(b^2 = a^2 - c^2\). Substitute \(a\) and \(c\) to find \(b^2\), then write the standard form of the ellipse equation as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis centered at the origin, it is (x²/a²) + (y²/b²) = 1, where 'a' is the distance from the center to a vertex, and 'b' relates to the minor axis length. Understanding this form helps translate geometric information into an algebraic equation.

Relationship Between Vertices, Foci, and Axes Lengths

In an ellipse, the vertices lie on the major axis at a distance 'a' from the center, while the foci lie at a distance 'c'. These distances satisfy the equation c² = a² - b², linking the focal distance, vertex distance, and minor axis length. Knowing this relationship allows calculation of missing parameters to write the ellipse equation.

Center and Orientation of the Ellipse

The center of the ellipse is the midpoint between the vertices and foci. The orientation (horizontal or vertical) is determined by the axis along which the vertices and foci lie. In this problem, both vertices and foci lie on the x-axis, indicating a horizontal major axis centered at the origin, which guides the form of the ellipse equation.