# How do you graph x^2/64-y^2=1 and identify the foci and asympototes?

May 27, 2017

For a hyperbola of the form:
${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1 \text{ [1]}$
The foci are at $\left(h \pm \sqrt{{a}^{2} + {b}^{2}} , k\right)$
The asymptotes are: $y = \pm \frac{b}{a} \left(x - h\right) + k$

#### Explanation:

Matching the given equation,

${x}^{2} / 64 - {y}^{2} = 1 \text{ [2]} ,$

with equation [1], we observe that $a = 8 , b = 1 , \mathmr{and} h = k = 0$.

With these values the foci are at:

$\left(0 - \sqrt{{8}^{2} + {1}^{2}} , 0\right)$ and $\left(0 + \sqrt{{8}^{2} + {1}^{2}} , 0\right)$

Simplify:

$\left(- \sqrt{65} , 0\right)$ and $\left(\sqrt{65} , 0\right)$

The asymptotes are:

$y = - \frac{1}{8} \left(x - 0\right) + 0$ and $y = \frac{1}{8} \left(x - 0\right) + 0$

Simplify:

$y = - \frac{1}{8} x$ and $y = \frac{1}{8} x$

The graph is:

graph{(x-0)^2/8^2-(y-0)^2/1^2=1 [-10, 10, -5, 5]}