Question

An ellipse has foci at (–1, 1) and (–1, –7). The major axis is 12 units long. Show work for all answers.
-Determine the center of the ellipse.
-Determine the vertices of the ellipse.
-Find the equation of the ellipse.

Answer 1

only the "y-values" change with the foci , this tells you the major axis is vertical
general equation of vertical ellipse
\[\frac{(y-k)^{2}}{a^{2}} + \frac{(x-h)^{2}}{b^{2}} = 1\]
where a^2 = b^2 + c^2
the distance between foci is 1-(-7) = 8
2c = 8 --> c = 4 --> c^2 = 16
given
2a = 12 --> a = 6 --> a^2 = 36
solve for b
36 = b^2 +16
b = sqrt(20)

Now to find center, take midpoint of the foci
--> (-1,-3)
h = -1 , k = -3

\[\frac{(y+3)^{2}}{36}+\frac{(x+1)^{2}}{20} = 1\]