Question

Find the equation of the hyperbola with the following properties. Express your answer in standard form.Foci at (6, -8) and (6, -14)Vertices at (6, -10) and (6,-12)

Answer 1

Given

The foci are given (6, -8) and (6, -14).

The vertices are given (6, -10) and (6,-12).

Step-by-step explanation

To determine the equation of hyperbola ,

Use the general equation of hyperbola .

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

The distance between foci is denoted as 2c.

`√((6-6)^2+(-14+8)^2)=√(36)=6`
`\begin{gathered} 2c=6 \\ c=3 \end{gathered}`

The distance between vertices is denoted as 2a.

`√((6-6)^2+(-12+10)^2)=√(4)=2`
`\begin{gathered} 2a=2 \\ a=1 \end{gathered}`

Now find the value of b,

`\begin{gathered} c^2=a^2+b^2 \\ 3^2=1^2+b^2 \\ b^2=9-1 \\ b^2=8 \end{gathered}`