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}{80}-y^2 = 1\\), \\(\frac{x^2}{81}-\frac{y^2}{80} = 1\\), \\(\frac{x^2}{81}-\frac{y^2}{81} = 1\\), \\(x^2 - \frac{y^2}{80} = 1\\)

Explanation:

Step1: Find $b^2$ using hyperbola relation

For hyperbolas, $c^2 = a^2 + b^2$. Substitute $a=1$, $c=9$:
$$9^2 = 1^2 + b^2$$
$$81 = 1 + b^2$$
$$b^2 = 81 - 1 = 80$$

Step2: Write standard horizontal hyperbola equation

The standard form for a horizontal transverse axis hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Substitute $a^2=1$, $b^2=80$:
$$\frac{x^2}{1} - \frac{y^2}{80} = 1$$
$$x^2 - \frac{y^2}{80} = 1$$

Answer:

$\frac{x^2}{1} - \frac{y^2}{80} = 1$ (equivalent to $x^2 - \frac{y^2}{80} = 1$)