How do you write an equation for a hyperbola with vertices (1, 3) and (-5, 3), and foci (3, 3) and (-7, 3)?

Oct 30, 2016

The equation is:

${\left(x - - 2\right)}^{2} / {3}^{2} - {\left(y - 3\right)}^{2} / {4}^{2} = 1$

Explanation:

Please notice that the vertices are of the forms:

$\left(h - a , k\right)$ and $\left(h + a , k\right)$ specifically $\left(- 5 , 3\right)$ and $\left(1 , 3\right)$

The same information can be deduced from the foci, which have the forms:

$\left(h - c , k\right)$ and $\left(h + c , k\right)$ specifically $\left(- 7 , 3\right)$ and $\left(3 , 3\right)$

The standard form for the equation of a hyperbola, where the vertices and foci have these properties, is the horizontal transverse axis form:

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

$k = 3$ by observation:

${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - 3\right)}^{2} / {b}^{2} = 1$

Compute h and a:

$- 5 = h - a$ and $1 = h + a$

$2 h = - 4$

$h = - 2$

$a = 3$

${\left(x - - 2\right)}^{2} / {3}^{2} - {\left(y - 3\right)}^{2} / {b}^{2} = 1$

To complete the equation, we only need the value of b but, to find the value of b, we must, first, find the value of c:

Using the $\left(h + c , k\right)$ form for the focus point, $\left(3 , 3\right)$, we substitute -2 for h, set the right side equal to 3, and then solve for c:

$- 2 + c = 3$

$c = 5$

Solve for b, using the equation ${c}^{2} = {a}^{2} + {b}^{2}$:

${5}^{2} = {3}^{2} + {b}^{2}$

$b = 4$

${\left(x - - 2\right)}^{2} / {3}^{2} - {\left(y - 3\right)}^{2} / {4}^{2} = 1$