# How do you write the equation of a hyperbola in standard form given Foci: (3,+-2) and Asymptotes: y = +-2(x-3)?

Oct 31, 2016

The equation of the hyperbola is ${\left(y\right)}^{2} / \left(\frac{16}{5}\right) - {\left(x - 3\right)}^{2} / \left(\frac{4}{5}\right) = 1$

#### Explanation:

The standard equation of a hyperbola is
${\left(y - k\right)}^{2} / {b}^{2} - {\left(x - h\right)}^{2} / {a}^{2} = 1$
The slopes of the asymptotes are $\pm \frac{b}{a}$
so $\pm \frac{b}{a} = \pm 2$ $\implies$$b = 2 a$
The foci are $h , k \pm c$
So $h = 3$
and $k + c = 2$ and $k - c = - 2$
So from the eqautions $k = 0$ and $c = 2$
${c}^{2} = {a}^{2} + {b}^{2}$
Therefore $4 = {a}^{2} + 4 {a}^{2} = 5 {a}^{2}$ $\implies$ ${a}^{2} = \frac{4}{5}$
And ${b}^{2} = 4 {a}^{2} = \frac{16}{5}$

So the equation of the hyperbola is
${\left(y\right)}^{2} / \left(\frac{16}{5}\right) - {\left(x - 3\right)}^{2} / \left(\frac{4}{5}\right) = 1$