# How do you find an equation that models a hyperbolic lens with a=12 inches and foci that are 26 inches apart, assume that the center of the hyperbola is the origin and the transverse axis is vertical?

##### 1 Answer
Apr 17, 2017

The process in described in the explanation.

#### Explanation:

Here is the equation of a hyperbola with a vertical transverse axis:

${\left(y - k\right)}^{2} / {a}^{2} - {\left(x - h\right)}^{2} / {b}^{2} = 1 \text{ [1]}$

We are given that the center is the origin; this means that $h = k = 0$

${\left(y - 0\right)}^{2} / {a}^{2} - {\left(x - 0\right)}^{2} / {b}^{2} = 1 \text{ [2]}$

We are given that $a = 12$:

${\left(y - 0\right)}^{2} / {12}^{2} - {\left(x - 0\right)}^{2} / {b}^{2} = 1 \text{ [2]}$

We are given that the foci are 26 inches apart; this means that the focal length is 13:

$13 = \sqrt{{a}^{2} + {b}^{2}}$

${13}^{2} = {12}^{2} + {b}^{2}$

$b = \sqrt{169 - 144}$

$b = \sqrt{25}$

$b = 5$

${\left(y - 0\right)}^{2} / {12}^{2} - {\left(x - 0\right)}^{2} / {5}^{2} = 1 \leftarrow$ answer