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Answer:
To find the equation of an ellipse with a vertical major axis, we need to determine the values of the center, the distance between the center and one of the covertices (c), and the distance between the center and one of the vertices (a).
Given:
Center: (-1, 3)
Distance between center and covertex: 4
Distance between center and vertex: 6
We can start by determining the values of c and a. For an ellipse, the relationship between c, a, and b (the distance between the center and one of the endpoints of the minor axis) is given by the equation:
c^2 = a^2 - b^2
Since the major axis is vertical, a represents the distance from the center to the vertex, and b represents the distance from the center to the covertex.
Using the given information, we have:
c = 4
a = 6
Let's calculate the value of b:
b^2 = a^2 - c^2
b^2 = 6^2 - 4^2
b^2 = 36 - 16
b^2 = 20
b = sqrt(20) = 2sqrt(5)
Now we have all the necessary values to write the equation of the ellipse. The standard form of the equation for an ellipse with a vertical major axis is:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Where (h, k) represents the coordinates of the center.
Substituting the values we found:
(x + 1)^2 / 36 + (y - 3)^2 / (2sqrt(5))^2 = 1
Simplifying further:
(x + 1)^2 / 36 + (y - 3)^2 / 20 = 1
Therefore, the equation of the ellipse with a vertical major axis, center (-1, 3), distance between the center and one of the covertices equal to 4, and the distance between the center and one of the vertices equal to 6 is:
(x + 1)^2 / 36 + (y - 3)^2 / 20 = 1