$$\require{cancel}$$
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \label{4.6.1}$
represents a cylinder of elliptical cross-section, whose axis coincides with the $$z$$-axis. The Equation to a cylinder with an axis in another position and with another orientation can be obtained by the usual processes of translation and rotation of axes (see Section 3.7).