Question

find the equation of the ellipse with vertices at (1,-2) and (1,8) and with a minor axis of length 6

Answer 1

Vertices: (1, -2) ; (1,8)

Length of Minor axis: 2b=6

Use the vertices to find the center of the ellipse:

$\begin{gathered} \frac{(x_1+x_2)}{2},\frac{(y_1+y_2)}{2} \\ \\ \frac{1+1}{2},\frac{-2+8}{2} \\ \\ \frac{2}{2},\frac{6}{2} \\ \\ (1,3) \end{gathered}$

And the length of the major axis:

$\begin{gathered} \sqrt[]{(x_1-x_2)^2+(y_1-y_2)^2} \\ \\ \sqrt[]{(1-1)^2+(-2-8)^2} \\ \\ =\sqrt[]{0+(-10)^2} \\ \\ =\sqrt[]{100} \\ 2a=10 \end{gathered}$

Then, you get the next equation:

$\begin{gathered} \text{Center (h,k)} \\ \frac{(x-h)^2}{a^2}+\frac{(y^2-k)^2}{b^2}=1 \\ \\ (h,k)=(1,3) \\ a=10/2=5 \\ b=6/2=3 \\ \\ \frac{(x-1)^2}{25}+\frac{(y^2-3)^2}{9}=1 \end{gathered}$