Chapter 8 - Section 8.4 - Plane Curves and Parametric Equations - 8.4 Exercises - Page 619: 57

$\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ Equation of a hyperbola with center at origin and vertical transverse axis.

$x=a~tan~θ$ $tan~θ=\frac{x}{a}$ $y=b~sec~θ$ $sec~θ=\frac{y}{b}$ $sec^2θ=1+tan^2θ$ $\frac{y^2}{b^2}=1+\frac{x^2}{a^2}$ $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$