Question

what is the equation of a hyperbola in standard form when c = 13 and a = 5? assume that the transverse axis is vertical.
a. $dfrac{y^2}{25} - dfrac{x^2}{144} = 1$
b. $dfrac{y^2}{144} - dfrac{x^2}{25} = 1$
c. $dfrac{x^2}{25} - dfrac{y^2}{144} = 1$
d. $dfrac{x^2}{144} - dfrac{y^2}{25} = 1$

Explanation:

Step1: Find $b^2$ via hyperbola identity

For hyperbolas, $c^2 = a^2 + b^2$. Substitute $c=13$, $a=5$:
$13^2 = 5^2 + b^2$
$169 = 25 + b^2$
$b^2 = 169 - 25 = 144$

Step2: Use vertical transverse axis form

Standard form for vertical transverse axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
Substitute $a^2=25$, $b^2=144$:
$\frac{y^2}{25} - \frac{x^2}{144} = 1$

Answer:

A. $\frac{y^2}{25} - \frac{x^2}{144} = 1$