Final answer:
To find the equation of the ellipse, we first determine the lengths of the semi-major axis (a) and semi-minor axis (b) from the vertices and co-vertices. The equation is derived from the general formula for an ellipse with a vertical major axis, which is (x-h)^2/b^2 + (y-k)^2/a^2 = 1. Substituting the values, we get the specific equation for the ellipse: (x+2)^2/9 + (y-6)^2/16 = 1.
Step-by-step explanation:
The question asks to write an equation of an ellipse with given vertices and co-vertices. The vertices at (-2, 2) and (-2, 10) suggest that the major axis is vertical, while the co-vertices at (-5,6) and (1,6) indicate a horizontal minor axis, centered at (-2, 6). The distance between the vertices along the major axis is 8 units (from 2 to 10), which means the semi-major axis length, a, is 4 units. The distance between the co-vertices along the minor axis is 6 units (from -5 to 1), so the semi-minor axis length, b, is 3 units.
The general equation for an ellipse centered at (h, k) with a vertical major axis is (x-h)^2/b^2 + (y-k)^2/a^2 = 1. Substituting the values for h, k, a, and b into the equation, we get:
(x+2)^2/3^2 + (y-6)^2/4^2 = 1
Which simplifies to:
(x+2)^2/9 + (y-6)^2/16 = 1
This is the equation of the ellipse with the given vertices and co-vertices.