1.34 Hyperbolic area

(iii) For the hyperbola, we substitute $x=a\cosh u$ in

Consider the geometric meaning of the change of variables for the hyperbola $x^{2}-y^{2}=a^{2}$. The hyperbola has centre $O=(0,0)$ and vertex $A=(a,0)$; the lines $y=\pm x$ are asymptotes to the hyperbola as $x\rightarrow\pm\infty$. We ‘parametrize’ the hyperbola by introducing a new variable $u$ so that $x=a\cosh u$ and $y^{2}=x^{2}-a^{2}=a^{2}(\cosh^{2}u-1)=a^{2}\sinh^{2}u$. Thus $(a\cosh u,a\sinh u)$ is a typical point on the right branch of the hyperbola.

Let $P=(a\cosh v,a\sinh v)$ be a point on the hyperbola and let $Q$ be its projection to the $x$-axis.