Question

1. what is the equation of a hyperbola with a = 7 and c = 9? assume that the transverse axis is horizontal

options:
$\frac{x^2}{49} - \frac{y^2}{81} = 1$
$\frac{x^2}{81} - \frac{y^2}{49} = 1$
$\frac{x^2}{81} - \frac{y^2}{32} = 1$
$\frac{x^2}{49} - \frac{y^2}{32} = 1$

Explanation:

Step1: Recall hyperbola relation

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

Step2: Calculate $b^2$

Substitute $a=7$, $c=9$:
$b^2 = c^2 - a^2 = 9^2 - 7^2 = 81 - 49 = 32$

Step3: Write standard equation

Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Substitute $a^2=49$, $b^2=32$:
$\frac{x^2}{49} - \frac{y^2}{32} = 1$

Answer:

$\boldsymbol{\frac{x^2}{49} - \frac{y^2}{32} = 1}$