Question

1. what is the equation of a hyperbola with a = 8 and c = 20? assume that the transverse axis is horizontal.

options:
\\(\frac{x^2}{64} - \frac{y^2}{336} = 1\\)
\\(\frac{x^2}{8} - \frac{y^2}{20} = 1\\)
\\(\frac{x^2}{64} - \frac{y^2}{20} = 1\\)
\\(\frac{x^2}{64} - \frac{y^2}{200} = 1\\)

Explanation:

Step1: Recall hyperbola relation

$c^2 = a^2 + b^2$

Step2: Substitute given values

$20^2 = 8^2 + b^2$

Step3: Calculate $b^2$

$b^2 = 400 - 64 = 336$

Step4: Write standard equation

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

Answer:

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