Question

1. what is the equation of a hyperbola with a = 6 and c = 10? assume that the transverse axis is horizontal. options: \\(\frac{x^2}{6} - \frac{y^2}{10} = 1\\), \\(\frac{x^2}{36} - \frac{y^2}{64} = 1\\), \\(\frac{x^2}{64} - \frac{y^2}{36} = 1\\), \\(\frac{x^2}{36} - \frac{y^2}{100} = 1\\)

Explanation:

Step1: Recall hyperbola relationship

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

Step2: Calculate $b^2$

Substitute $a=6$, $c=10$:
$b^2 = c^2 - a^2 = 10^2 - 6^2 = 100 - 36 = 64$

Step3: Write standard hyperbola equation

Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Substitute $a^2=36$, $b^2=64$:
$\frac{x^2}{36} - \frac{y^2}{64} = 1$

Answer:

$\frac{x^2}{36} - \frac{y^2}{64} = 1$ (the third option)