# How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola x^2/9-y^2/25=1?

Dec 26, 2017

vertices $\left(3 , 0\right) \mathmr{and} \left(- 3.0\right)$
Foci $\left(\sqrt{34} , 0\right) \mathmr{and} \left(- \sqrt{34} , 0\right)$
Aymptotes $y = \frac{5}{3} x \mathmr{and} y = - \frac{5}{3} x$

#### Explanation:

Given -

${x}^{2} / 9 - {y}^{2} / 25 = 1$

This hyperbola equation is in the form

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

If this is the case then

Its vertices are $\left(a , 0\right) \mathmr{and} \left(- a , 0\right)$
Its foci are $\left(c , 0\right) \mathmr{and} \left(- c , 0\right)$

Its asymptotes are $y = \frac{b}{a} x \mathmr{and} y = - \frac{b}{a} x$

Then we have to find the values of $a , b \mathmr{and} c$ from the given equation.

$a = \sqrt{9} = 3$
$b = \sqrt{25} = 5$
${c}^{2} = {a}^{2} + {b}^{2}$
$c = \pm \sqrt{9 + 25} = \pm \sqrt{34}$

Then
vertices $\left(3 , 0\right) \mathmr{and} \left(- 3.0\right)$
Foci $\left(\sqrt{34} , 0\right) \mathmr{and} \left(- \sqrt{34} , 0\right)$
Aymptotes $y = \frac{5}{3} x \mathmr{and} y = - \frac{5}{3} x$