Chapter 1.7, Problem 12E

(a)

To determine

To find:The Cartesian equation of the curve.

Explanation of Solution

Given: The parametric equations are $x={\tan }^2\theta$ , $y=\sec \theta$

Consider the expression.

  $\begin{array}{c}x={\tan }^2\theta \\ x={\sec }^2\theta -1\\ x+1={\sec }^2\theta \\ \sec \theta =\sqrt{x+1}\end{array}$

And,

  $\begin{array}{c}y=\sec \theta \\ =\sqrt{x+1}\end{array}$

Therefore, the equation in Cartesian form is $y=\sqrt{x+1}$ .

(b)

To determine

To sketch:The curve by using the parametric equation.

Explanation of Solution

Given: The parametric equations are $x={\tan }^2\theta$ , $y=\sec \theta$

The sketch of the graph is shown in figure below.

  

Figure (1)

Therefore, the required graph is shown in figure (1).