# 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?

Apr 17, 2017

The process in described in the explanation.

#### Explanation:

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

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

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

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

We are given that $a = 12$:

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

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