Question

Please help.medals!!!
Write an equation of the hyperbola given that the center is at (2, -3), the vertices are at (2, 3) and
(2, - 9), and the foci are at (2, ± 2√10).

Answer 1

Starting with the general form of a vertical hyperbola (x term is negative & y term positive):
\[1=\frac{ (y-k) ^{2}}{ a }-\frac{ (x-h) ^{2}}{ b } \]
Where the center is (h,k)
The vertices are at (h,k+a) & (h,k-a)
And the foci are located at (h,k+c) & (h,k-c) where:
\[c^{2}=a^{2}+b^{2}\]
Clearly from the problem we know h=2 & k=-3 and with a little reverse engineering:
(2,3)=(2,-3+6) & (2,-9)=(2,-3-6) which shows that a = 6.
Since we know the foci explicitly we solve for them in much the same way:
\[k \pm c = \pm 2\sqrt{10} \ \ \ \ \ \rightarrow \ \ \ \ \ \ c=\mp (3 \pm 2\ \sqrt{10}) \]
At this point, I have to stop and ask you to check the problem again to make sure the values listed were correct because carrying on with my calculation I ended up with some unexpected and (presumably) incorrect results. Granted it has been a while since I have done one of these calculations so before I go any further in trying to find an error (that may not even be there) please check that the foci values listed were correct and let me know in the comments or pm me. Thank you.
If you want some pictures to help in visualization might I suggest:
http://sites.csn.edu/istewart/mathweb/Math127/hyperbolas/hyperbolas.htm