Find the standard form of the equation of each ellipse and give the location of its foci.

Verified step by step guidance

1. Identify the center of the ellipse. Since the vertices and foci are symmetric about the origin, the center is at (0, 0).
2. Determine the lengths of the major and minor axes. The vertices are at (6, 0) and (-6, 0), so the length of the major axis is 12, and the semi-major axis length is \(a = 6\). The foci are at (0, 10) and (0, -10), so the distance from the center to each focus is \(c = 10\).
3. Since the vertices lie on the x-axis and the foci lie on the y-axis, the major axis is vertical. This means the standard form of the ellipse equation is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), where \(a > b\) and \(a\) corresponds to the vertical axis.
4. Use the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^2 = a^2 - b^2\). Substitute \(a = 10\) and \(c = 6\) (note the correction: since the foci are at (0, ±10), \(c=10\), and vertices at (±6, 0), \(a=6\); but since the major axis is vertical, \(a=10\) and \(c=6\)). Solve for \(b^2\).
5. Write the standard form equation of the ellipse using the values of \(a^2\) and \(b^2\), and list the coordinates of the foci as given.

Key Concepts

Standard Form of an Ellipse Equation

The standard form of an ellipse equation depends on the orientation of its major axis. For a vertical major axis centered at the origin, the equation is (x^2 / b^2) + (y^2 / a^2) = 1, where a > b. Here, 'a' is the distance from the center to a vertex along the major axis, and 'b' is the distance along the minor axis.

Vertices and Axes of an Ellipse

Vertices are points where the ellipse intersects its major axis, representing the longest diameter. The major axis is the line segment through the center connecting the vertices, while the minor axis is perpendicular to it and shorter. Identifying vertices helps determine 'a' and 'b' values for the ellipse equation.

Foci and the Relationship Between a, b, and c

The foci are two fixed points inside the ellipse used to define it. The distance from the center to each focus is 'c', related to 'a' and 'b' by the equation c^2 = a^2 - b^2. Knowing the foci locations helps verify the ellipse's dimensions and is essential for finding the standard form.