Find an equation of an ellipse satisfying the given conditions:
Vertices: (-1, -8) and (-1, 4); and the length of the minor axis is 10.
the major axis is vertical, and
a = 5, b = 6 , and centre is (-1,-2)
equation:
(x+1)^2/25 + (y+2)^2/36 = 1
To find the equation of an ellipse, we need to use the standard form equation for an ellipse. The general equation for an ellipse with its center at the point (h, k) is:
(x - h)²/a² + (y - k)²/b² = 1
where (h, k) is the center of the ellipse, a is the semi-major axis, b is the semi-minor axis.
Given the vertices: (-1, -8) and (-1, 4), we can determine the center of the ellipse, (h, k), which is simply the midpoint of the vertices. Since the x-coordinate of both vertices is the same (-1), the center of the ellipse has an x-coordinate of -1 as well. To find the y-coordinate of the center, we take the average of the y-coordinates of the vertices:
k = (-8 + 4)/2 = -4/2 = -2
So, the center of the ellipse is (-1, -2).
Next, we need to determine the semi-major axis, a. The length of the major axis is the distance between the two vertices, which in this case is 4 - (-8) = 12. Since the major axis is twice the length of the semi-major axis, we have:
a = 12/2 = 6
We're also given that the length of the minor axis is 10. Since the minor axis is twice the length of the semi-minor axis, we have:
b = 10/2 = 5
Now, we can substitute the values of (h, k), a, and b into the standard form equation:
(x - (-1))²/6² + (y - (-2))²/5² = 1
Simplifying this equation, we get:
(x + 1)²/36 + (y + 2)²/25 = 1
So, the equation of the ellipse satisfying the given conditions is:
(x + 1)²/36 + (y + 2)²/25 = 1