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 by finding the midpoint between the vertices. The vertices are at (-11, 3) and (15, 3), so the center is at \(\left( \frac{-11 + 15}{2}, \frac{3 + 3}{2} \right)\).
2. Calculate the lengths of the major and minor axes. The distance between the vertices gives the length of the major axis, so find \$2a = 15 - (-11)\(, then solve for \)a\(. The distance between the co-vertices (2, 8) and (2, -2) gives \)2b\(, so find \)b$ similarly.
3. Write the standard form of the ellipse equation. Since the major axis is horizontal (because the vertices differ in the x-coordinate), the equation is \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center.
4. Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\) for ellipses with a horizontal major axis.
5. Determine the coordinates of the foci by moving \(c\) units left and right from the center along the major axis, resulting in foci at \((h - c, k)\) and \((h + c, k)\).

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, the form is ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1, where (h, k) is the center, a is the semi-major axis length, and b is the semi-minor axis length. This form helps identify the ellipse's size and position on the coordinate plane.

Center, Vertices, and Axes of an Ellipse

The center of the ellipse is the midpoint between its vertices. The vertices lie along the major axis and determine the length of the ellipse's longest diameter (2a). The minor axis is perpendicular to the major axis and has length 2b. Identifying these points from the graph is essential to write the ellipse equation.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations are found using c^2 = a^2 - b^2, where c is the distance from the center to each focus. Knowing the foci helps understand the ellipse's geometric properties and is required in the problem.