Final answer:
The equation for the ellipse with given vertices and a point on it is ((x+5)²/5²) + ((y-6)²/8²) = 1, where the vertices provide the length of the major axis and the point gives the length of the semi-minor axis.
Step-by-step explanation:
Given the two vertices of the ellipse at (-5,-2) and (-5,14), we can find the length of the major axis. Since the vertices are on a vertical line, this implies the major axis is vertical. The distance between the two vertices is the length of the major axis (2a), which is 16 units (14 - (-2)).
Therefore, the semi-major axis (a) is 8 units. The center of the ellipse is at (-5,6), which is the midpoint between the two vertices.
Since the point (0,6) also lies on the ellipse and has the same y-coordinate as the center, it lies on the horizontal line passing through the center of the ellipse. This means (0,6) is on the semi-minor axis (b). We can calculate the distance from the center to this point to find the semi-minor axis length, which turns out to be 5 units (0 - (-5)).
The standard form of the equation of an ellipse with vertical major axis is
(x-h)²/b² + (y-k)²/a² = 1, where (h, k) is the center of the ellipse. Substituting the values we have, the equation of the ellipse on which these points lie is:
((x+5)²/5²) + ((y-6)²/8²) = 1