Chapter 10.7, Problem 10AYU

To determine

The graph of the curve of a parametric equation $x\left(t\right)=\sqrt{2t},y\left(t\right)=4t;t\ge 0$, show its orientation. Find the rectangular equation of curve.

Answer to Problem 10AYU

Solution:

The graph of the parametric equations $x\left(t\right)=\sqrt{2t},y\left(t\right)=4t;t\ge 0$ is

The rectangular equation of the parametric equation $x\left(t\right)=\sqrt{2t},y\left(t\right)=4t;t\ge 0$ is $2x^2-y=0$.

Explanation of Solution

Given information:

The parametric equation, $x\left(t\right)=\sqrt{2t},y\left(t\right)=4t;t\ge 0$

Explanation:

To draw the given parametric equation, plug some values of $t$ in the given equation and find few points on the curve.

The interval for $t$ is $t\ge 0$

$\begin{array}{cccc}t& x\left(t\right)=\sqrt{2t}& y\left(t\right)=4t& \left(x,y\right)\\ 0& 0& 0& \left(0,0\right)\\ 1& \sqrt{2}& 4& \left(\sqrt{2},4\right)\\ 2& 2& 8& \left(2,8\right)\\ 3& \sqrt{6}& 12& \left(\sqrt{6},12\right)\\ 4& \sqrt{8}=2\sqrt{2}& 16& \left(2\sqrt{2},16\right)\end{array}$

The orientation is the curve traced out in a certain direction by the corresponding succession of points $\left(x,y\right)$.

The arrow shows the orientation along the curve as $t$ varies from $t\ge 0$

As $t$ takes values from $t\ge 0$, the corresponding orientation is from $\left(0,0\right)$ to $\left(2\sqrt{2},16\right)$.

Use points in the table to sketch the curve.

The graph of the parametric equation $x\left(t\right)=\sqrt{2t},y\left(t\right)=4t;t\ge 0$ is

Let $x=\sqrt{2t}$ and $y=4t$

$\Rightarrow x^2=2t$ and $y=4t$

$\Rightarrow 2x^2=4t$ and $y=4t$

$\Rightarrow 2x^2=y$

$\Rightarrow 2x^2-y=0$

The rectangular equation of the parametric equation $x\left(t\right)=\sqrt{2t},y\left(t\right)=4t;t\ge 0$ is $2x^2-y=0$.