# How do you write the equation of a hyperbola given vertices (-5, 0) and (5, 0) and foci (-7, 0) and (7, 0)?

Feb 25, 2018

${x}^{2} / 25 - {y}^{2} / 24 = 1$.

#### Explanation:

We know that, for the Hyperbola $S : {x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = 1$,

the Focii and the Vertices are $\left(\pm a e , 0\right) \mathmr{and} \left(\pm a , 0\right)$, resp.

Here, $e$, the Eccentricity of $S$, is given by, ${b}^{2} = {a}^{2} \left({e}^{2} - 1\right)$.

Clearly, in our case, $a = 5 , a e = 7 , \text{ so that, } e = \frac{7}{a} = \frac{7}{5}$.

$\therefore {b}^{2} = {a}^{2} \left({e}^{2} - 1\right) = 25 \left(\frac{49}{25} - 1\right) = 24$.

Hence follows the eqn. of $S : {x}^{2} / 25 - {y}^{2} / 24 = 1$.