# How do you graph y^2/100-x^2/75=1 and identify the foci and asympototes?

Nov 11, 2016

The foci are F$= \left(0 , 5 \sqrt{7}\right)$ and F'$= \left(0 , - 5 \sqrt{7}\right)$
The asymptotes are $y = \frac{2 x}{\sqrt{3}}$ and $y = - \frac{2 x}{\sqrt{3}}$

#### Explanation:

This is an up-down hyperbola.
The general equation is ${y}^{2} / {a}^{2} - {x}^{2} / {b}^{2} = 1$
The center is $\left(0 , 0\right)$
To determine the foci, we calculate $c = \sqrt{{a}^{2} + {b}^{2}} = \pm \sqrt{175} = \pm 5 \sqrt{7}$
The foci are F$= \left(0 , + 5 \sqrt{7}\right)$ and F'$= \left(0 , - 5 \sqrt{7}\right)$

The vertices are $\left(0 , 10\right)$ and $\left(0 , - 10\right)$

The slopes of the asymptotes are $\pm \frac{10}{\sqrt{75}} = \pm \frac{10}{5 \sqrt{3}} = \pm \frac{2}{\sqrt{3}}$
The equations of the asymptotes are $y = \frac{2 x}{\sqrt{3}}$ and $- \frac{2 x}{\sqrt{3}}$
graph{((y^2/100)-(x^2/75)-1)(y-(2x/sqrt3))(y+(2x/sqrt3))=0 [-74, 74.1, -37.07, 37.06]}