Hyperbola Ebola

Since the center, focus, and vertex all lie on the line \(y=3\), the hyperbola opens left/right. This means that we use this form:

\({(x-h)^2\over a^2}-{(y-k)^2\over b^2}=1\)

Now, we substitute the values as follows:

     1. h is the x-coordinate of the center

     2. k is the y-coordinate of the center

     3. a is the distance from the center to the vertex

     4. b² = c² - a², wherein c is the distance from the center to the focus

Which means:

     1. \(h=3\)

     2. \(k=3\)

     3. \(a=\sqrt{(3-6)^2+(3-3)^3}=3\)

     4. \(c=\sqrt{(3-8)^2+(3-3)^2}=5\), \(b^2=25-9=16\)

Substituting, we get \({(x-3)^2\over9}-{(y-3)^2\over16}=1\), and that is your equation.