Question

1. - / 0.81 points

consider the following.
$x = \tan^{2}(\theta)$, $y = \sec(\theta)$, $-\pi/2 < \theta < \pi/2$
(a) eliminate the parameter to find a cartesian equation of the curve.
(b) sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.

Explanation:

Step1: Recall the trigonometric identity

We know the identity \(1+\tan^{2}\theta=\sec^{2}\theta\).

Step2: Substitute the given parametric equations

Given \(x = \tan^{2}\theta\) and \(y=\sec\theta\). Substitute into the identity: \(1 + x=y^{2}\).

Step3: Consider the domain of \(y\)

Since \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\), \(\sec\theta=y> 0\).

Answer:

\(y^{2}-x = 1,y>0\)