# How do you write an equation of a hyperbola given the foci of the hyperbola are (8 , 0) and (−8 , 0), and the asymptotes are y =sqrt(7)x and y =−sqrt(7)x?

In many textbooks, standard notation for such a hyperbola is ${\left(\frac{x}{a}\right)}^{2} - {\left(\frac{y}{b}\right)}^{2} = 1$. For such a hyperbola, the asymptotes are $y = \setminus \pm \setminus \frac{b}{a} x$. Hence, $\frac{b}{a} = \setminus \sqrt{7}$. Also for such a hyperbola, the foci are at the points whose $x$ coordinates are $x = \setminus \pm \setminus \sqrt{{a}^{2} + {b}^{2}}$, implying that $\setminus \sqrt{{a}^{2} + {b}^{2}} = 8$ so that ${a}^{2} + {b}^{2} = 64$.
Subtituting $b = a \setminus \sqrt{7}$ into this last equation gives ${a}^{2} + 7 {a}^{2} = 64$, or ${a}^{2} = 8$ so that $a = \setminus \sqrt{8} = 2 \setminus \sqrt{2}$. This then implies that $b = 2 \setminus \sqrt{2} \setminus \sqrt{7} = 2 \setminus \sqrt{14}$.
The equation of the hyperbola is ${\left(\frac{x}{2 \setminus \sqrt{2}}\right)}^{2} - {\left(\frac{y}{2 \setminus \sqrt{14}}\right)}^{2} = 1$, which can also be written as ${x}^{2} / 8 - {y}^{2} / 56 = 1$ or $7 {x}^{2} - {y}^{2} = 56$.