# How do you find all the critical points to graph x^2/25 - y^2/49=1 including vertices?

Apr 15, 2018

Below

#### Explanation:

general form of a hyperbola: ${x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = 1$

${a}^{2} = 25$ so $a = 5$

${b}^{2} = 49$ so $b = 7$

the eccentricity is equal to ${b}^{2} = {a}^{2} \left({e}^{2} - 1\right)$

$49 = 25 \left({e}^{2} - 1\right)$

$\frac{49}{25} + 1 = {e}^{2}$

$\frac{74}{25} = {e}^{2}$

$e = \frac{\sqrt{74}}{5}$

Focus $\left(\pm a e , 0\right)$

$\left(\pm 5 \times \frac{\sqrt{74}}{5} , 0\right)$

$\left(\pm \sqrt{74} , 0\right)$

Directrix: $x = \frac{a}{e}$ SO $x = \frac{5}{\frac{\sqrt{74}}{5}}$ SO $x = \frac{25}{\sqrt{74}}$

Asymptote: $y = \frac{b}{a} x$ SO $y = \frac{7}{5} x$

Vertices: $\left(\pm 5 , 0\right)$