Chapter 12, Problem 5CLT

To determine

To graph: The curve represented by given parametric equation. Also to eliminate the parameter and write the rectangular equation.

Answer to Problem 5CLT

The required equation is $y=\frac{1}{2}{e}^{\frac{x}{2}}$ .

Explanation of Solution

Given data: $y$

The given parametric equations are $x=4\ln t$ and $y=\frac{1}{2}{t}^2$ .

Method/Formula used:

Substitute value of parameter from one variable to another.

Calculation:

The given parametric equations are

  $x=4\ln t$  … (1)

  $y=\frac{1}{2}{t}^2$  … (2)

From equation (1) $x=4\ln t\Rightarrow {\log }_{e}t=\frac{x}{4}$

  $\Rightarrow e^{\frac{x}{4}}=t$  … (3)

Substituting value of t in equation (2)

  $y=\frac{1}{2}{\left(e^{\frac{x}{4}}\right)}^2=\frac{1}{2}\left(e^{\frac{2x}{4}}\right)$$\because {\left(a^m\right)}^n=a^{mn}$

  $\Rightarrow y=\frac{1}{2}{e}^{\frac{x}{2}}$ .

Thus, the rectangular equation from the given parametric equation is $y=\frac{1}{2}{e}^{\frac{x}{2}}$ .

The graph of the parametric equation is the graph of the rectangular equation shown in Figure-1 here.