Question

4 what is the equation of a hyperbola with a = 8 and c = 20? assume that the transverse axis is horizontal. options: \\(\frac{x^2}{8} - \frac{y^2}{20} = 1\\), \\(\frac{x^2}{64} - \frac{y^2}{200} = 1\\), \\(\frac{x^2}{64} - \frac{y^2}{336} = 1\\), \\(\frac{x^2}{64} - \frac{y^2}{20} = 1\\)

Explanation:

Step1: Recall hyperbola relation

For hyperbolas, $c^2 = a^2 + b^2$

Step2: Substitute given values

$a=8$, $c=20$, so $20^2 = 8^2 + b^2$

Step3: Calculate $b^2$

$400 = 64 + b^2 \implies b^2 = 400 - 64 = 336$

Step4: Write standard equation

Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, substitute $a^2=64$, $b^2=336$
$\frac{x^2}{64} - \frac{y^2}{336} = 1$

Answer:

$\frac{x^2}{64} - \frac{y^2}{336} = 1$