Chapter 8 - Section 8.4 - Plane Curves and Parametric Equations - 8.4 Exercises - Page 617: 19

(a) See the graph. (b) $x^2-y^2=1,~~x\geq1,~~y\geq0$

(a) $t=0$: $x=sec~0=1$ $y=tan~0=0$ Point: $(1,0)$ $t=\frac{\pi}{4}$: $x=sec\frac{\pi}{4}=\sqrt 2$ $y=tan\frac{\pi}{4}=1$ Point: $(\sqrt 2,1)$ As $t→\frac{\pi}{4}$ both $x$ and $y$ $→∞$. So, $x\geq1$ and $y\geq0$ (b) $x=sec~t$ $x^2=sec^2t$ $y=tan~t$ $y^2=tan^2t$ $sec^2t=tan^2t+1$ $x^2=y^2+1$ $x^2-y^2=1$