Question

Find the standard equation of the hyperbola that has a vertex at (4, 2), focus at (4, 4) and a center at (4, -1).

Answer 1

I don't have much time, as I mentioned, but!

I will get you started by saying, you can apply much of what we applied in my previous answer, except, since the center isn't at (0, 0), a couple of things change. First off, the equation becomes:

\[-\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]

This is for a center (h, k). Note that this time it's - + instead of + -; that's because this hyperbola opens up and down instead of left and right (you can see that if you sketch it based on the parameters they give you above).

The other thing that changes is that the equation for the vertices is (h + a, k) and (h - a, k).