Question

In problem graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=\sqrt{2t},y=4t;t\ge 0$.

Short answer

The answer is$x=\sqrt{\frac{y}{2}.}$

Step-by-step solution

Step 1. Given information.

$x=\sqrt{t},y=4t\text{and}t\ge 0$is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$x=\sqrt{t},y=4tt\ge 0$

$t$	$x$	$y$	$(x,y)$
$0$	$0$	$0$	$(0,0)$
$1$	$1$	$4$	$(1,4)$
$2$	$\sqrt{2}$	$8$	$(\sqrt{2},8)$
$3$	$\sqrt{3}$	$12$	$(\sqrt{3},12)$

We find the rectangular equation by eliminating the parameter t from the parametric equations.

$$\begin{gathered} y=4t \\ t=\frac{y}{4} \end{gathered}$$

Put the value of t in the other equation.

$$\begin{gathered} x=\sqrt{2t} \\ x=\sqrt{2\times \frac{y}{4}} \\ x=\sqrt{\frac{y}{2}} \end{gathered}$$

Thus, the equation is$x=\sqrt{\frac{y}{2}.}$