Question

sketch the parametric curve by eliminating the parameter

$x=\tan t,y=\tan t,t\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

Short answer

The required graph is

Step-by-step solution

Given information

The parametric curve is$x=\tan t,y=\tan t,t\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

Calculation

Let us consider the parametric equations $x=\tan t,y=\tan t,t\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

The objective is to sketch the parametric curve by eliminating the parameters.

By equating the two parametric equations, the parameter $t$is eliminated.

Thus,

$$\begin{gathered} x=\tan t \\ x=y\text{since}y=\tan t \\ y=x \end{gathered}$$

To draw the graph of the equation, assume .$x=-2,-1,0,1,2$

Substitute $x=-2$in the equation $x=y.$

Then,

$$\begin{gathered} y=-2 \\ (x,y)=(-2,-2) \end{gathered}$$

Substitute $x=-1$in the equation $x=y$.

Then,

$$\begin{gathered} y=-1 \\ (x,y)=(-1,-1) \end{gathered}$$

Substitute $x=0$in the equation $x=y.$

Then,

$$\begin{gathered} y=0 \\ (x,y)=(0,0) \end{gathered}$$

Substitute $x=1$in the equation $x=y$. Then,

$$\begin{gathered} y=1 \\ (x,y)=(1,1) \end{gathered}$$

Further calculation

Substitute $x=2$in the equation $x=y$. Then,

$$\begin{gathered} y=2 \\ (x,y)=(2,2) \end{gathered}$$

The graphical representation by using the points $(-2,-2)(-1,-1)(0,0)(1,1)(2,2)$is as follows,

Therefore, the required equation after eliminating the parameter is $y=x$.