Find an equation of the ellipse having a major axis of lengt

Find an equation of the ellipse having a major axis of length 10 and foci at (-4, 0) and (-4, -8).

Solution

The equation of ellipse is given by

(x-h)^2/b^2+(y-k)^2/a^2 = 1, a>b, (h,k) being the (x,y) coordinates of the center.

for a vertical major axis.

For given equation, from the foci (-4, 0) and (-4, -8)
major axis: vertical
y-coordinate of center is -4 (midway between foci)
x-coordinate of center is -4

Hence center is at (-4, -4)

Length of major axis = 2a = 10

=> a = 5

Distance from center to focus is 4. Hence c = 4

Therefore b^2 = a^2 - c^2 = 25 - 16 = 9

b = 3

Hence equation of ellipse is

(x+4)^2/9 + (y+4)^2/25 = 1