How do you write the equation of the hyperbola given Foci: (-4,0),(4,0) and vertices (-1,0), (1,0)?

Jan 15, 2017

${x}^{2} - {y}^{2} / 15 = 1$

Explanation:

As focii $\left(- 4 , 0\right)$, $\left(4 , 0\right)$ and vertices $\left(- 1 , 0\right)$, $\left(1 , 0\right)$ lie on the same line $y = 0$, i.e. $x$-axis,

Further, as the mid point of vertices is $\left(0 , 0\right)$, the equation i of the type

${x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = 1$

As the distance between focii is $8$ and between vertices is $2$,

we have $c = \frac{8}{2} = 4$ and $a = \frac{2}{2} = 1$

and hence as ${c}^{2} = {a}^{2} + {b}^{2}$, $b = \sqrt{{4}^{2} - {1}^{2}} = \sqrt{15}$

and equation of hyperbola is

${x}^{2} / 1 - {y}^{2} / 15 = 1$ or $15 {x}^{2} - {y}^{2} = 15$
graph{15x^2-y^2-15=0 [-10, 10, -5, 5]}