Question

1. what is the equation of a hyperbola with a = 1 and c = 9? assume that the transverse axis is horizontal.

options:
$\frac{x^2}{81} - \frac{y^2}{80} = 1$
$x^2 - \frac{y^2}{81} = 1$
$\frac{x^2}{80} - y^2 = 1$
$x^2 - \frac{y^2}{80} = 1$

Explanation:

Step1: Recall hyperbola relation

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

Step2: Solve for $b^2$

Substitute $a=1$, $c=9$:
$b^2 = c^2 - a^2 = 9^2 - 1^2 = 81 - 1 = 80$

Step3: Write standard equation

Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Substitute $a^2=1$, $b^2=80$:
$\frac{x^2}{1} - \frac{y^2}{80} = 1$ or $x^2 - \frac{y^2}{80} = 1$

Answer:

$\boldsymbol{x^2 - \frac{y^2}{80} = 1}$ (corresponding to the last option)