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} ((x_1+x_2))/(2),((y_1+y_2))/(2) \\ \\ (1+1)/(2),(-2+8)/(2) \\ \\ (2)/(2),(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)} \\ ((x-h)^2)/(a^2)+((y^2-k)^2)/(b^2)=1 \\ \\ (h,k)=(1,3) \\ a=10/2=5 \\ b=6/2=3 \\ \\ ((x-1)^2)/(25)+((y^2-3)^2)/(9)=1 \end{gathered}`