Answer
(a) $x = 3~tan~t$
$y = 2~sec~t$
We can see the graph below.
(b) $\frac{y^2}{4} - \frac{x^2}{9} = 1$
Work Step by Step
(a) $x = 3~tan~t$
$y = 2~sec~t$
When $t = -\frac{3\pi}{8}$:
$x = 3~tan~\frac{-3\pi}{8} = -7.24$
$y = 2~sec~\frac{-3\pi}{8} = 5.23$
When $t = -\frac{\pi}{3}$:
$x = 3~tan~\frac{-\pi}{3} = -5.20$
$y = 2~sec~\frac{-\pi}{3} = 4$
When $t = -\frac{\pi}{4}$:
$x = 3~tan~\frac{-\pi}{4} = -3$
$y = 2~sec~\frac{-\pi}{4} = 2.83$
When $t = -\frac{\pi}{6}$:
$x = 3~tan~\frac{-\pi}{6} = 1.73$
$y = 2~sec~\frac{-\pi}{6} = 2.31$
When $t = 0$:
$x = 3~tan~0 = 0$
$y = 2~sec~0 = 2$
When $t = \frac{\pi}{6}$:
$x = 3~tan~\frac{\pi}{6} = 1.73$
$y = 2~sec~\frac{\pi}{6} = 2.31$
When $t = \frac{\pi}{4}$:
$x = 3~tan~\frac{\pi}{4} = 3$
$y = 2~sec~\frac{\pi}{4} = 2.83$
When $t = \frac{\pi}{3}$:
$x = 3~tan~\frac{\pi}{3} = 5.20$
$y = 2~sec~\frac{\pi}{3} = 4$
When $t = \frac{3\pi}{8}$:
$x = 3~tan~\frac{3\pi}{8} = 7.24$
$y = 2~sec~\frac{3\pi}{8} = 5.23$
We can see the graph below.
(b) $x = 3~tan~t$
$tan~t = \frac{x}{3}$
$tan^2~t = \frac{x^2}{9}$
$y = 2~sec~t$
$sec~t = \frac{y}{2}$
$sec^2~t = \frac{y^2}{4}$
$sec^2~t - tan^2~t = 1$
$\frac{y^2}{4} - \frac{x^2}{9} = 1$
