# Graph a Hyperbola with vertices(2,4),(8,4), foci(-2,4), (12,4), center(5,4). Can someone help me find the equation for this?

Mar 15, 2018

generalize form of equation of hyperbola is ${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1$

Where,$\left(h , k\right)$ is the coordinate of the centre,

and, $a$ is distance between the centre and the vertices,and $c$ is the distance between the centre and the focus.

and, ${a}^{2} + {b}^{2} = {c}^{2}$

So,given, $\left(h , k\right) = \left(5 , 4\right)$

$a = \sqrt{{\left(5 - 2\right)}^{2} + {\left(4 - 4\right)}^{2}} = 3$

$c = \sqrt{{\left(5 - 12\right)}^{2} + {\left(4 - 4\right)}^{2}} = 7$

so,${b}^{2} = \left({7}^{2} - {3}^{2}\right) = 40$

So,the equation of the hyperbola becomes, ${\left(x - 5\right)}^{2} / 9 - {\left(y - 4\right)}^{2} / 40 = 1$

Now,see the graph below

graph{((x-5)^2)/9 - ((y-4)^2)/40=1 [-20, 20, -10, 10]}