Question

Find an equation of the hyperbola having foci at (4, 8) and (4, 12) and vertices at (4,9) and (4, 11).

Answer 1

Since the x-coordinate of the vertices are constant, we can conclude it is a vertical hyperbola. The equation for a vertical hyperbola is given by:

`((y-k)^2)/(a^2)-((x-h)^2)/(b^2)=1`

Where:

`\begin{gathered} F=(h,k\pm c) \\ V=(h,k\pm a) \end{gathered}`

so:

`\begin{gathered} k+c=12 \\ so: \\ k=12-c \\ k-c=8 \\ 12-c-c=8 \\ 12-2c=8 \\ c=2 \\ k=10 \end{gathered}`
`\begin{gathered} k-a=9 \\ a=k-9 \\ a=1 \\ \end{gathered}`

Therefore:

`\begin{gathered} b=√(2^2-1^2) \\ b=√(3) \end{gathered}`

So, the equation is:

`((y-10)^2)/(1)-((x-4)^2)/(3)=1`