# How do you write the equation of a hyperbola given foci at (1, 5) and (7, 5) and with vertices at (2, 5) and (6, 5)?

Sep 6, 2016

Use the equation of hyperbola and ${a}^{2} + {b}^{2} = {c}^{2}$

#### Explanation:

The foci and the vertices all lie along the horizontal line y=5, so the hyperbola opens left/right. This implies that the x term goes first and the y term is subtracted.

The center is half way between the vertices at (4.5) = (h,k).

The vertices are 2 units from the center, so a=2. ${a}^{2} = 4$.

The foci are 3 units from center, so c =3.

Using ${a}^{2} + {b}^{2} = {c}^{2}$ gives ${b}^{2} = 5$.

Plugging into $\frac{{\left(x - h\right)}^{2}}{a} ^ 2 - \frac{{\left(y - k\right)}^{2}}{b} ^ 2 = 1$

gives $\frac{{\left(x - 4\right)}^{2}}{4} - \frac{{\left(y - 5\right)}^{2}}{5} = 1$