Chapter 10.7, Problem 20AYU

To determine

The graph of the curve of the parametric equation $x=2\cos t,y=3\sin t;0\le t\le \pi$, show its orientation. Find the rectangular equation of the curve.

Answer to Problem 20AYU

Solution:

The graph of the parametric equations $x=2\cos t,y=3\sin t;0\le t\le \pi$ is

The rectangular equation of the parametric equation $x=2\cos t,y=3\sin t;0\le t\le \pi$ is $\frac{x^2}{4}+\frac{y^2}{9}=1,y\ge 0$.

Explanation of Solution

Given information:

The parametric equation is $x=2\cos t,y=3\sin t;0\le t\le \pi$.

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 $0\le t\le \pi$.

$\begin{array}{cccc}t& x\left(t\right)& y\left(t\right)& \left(x,y\right)\\ 0& 2& 0& \left(2,0\right)\\ \frac{\pi }{4}& \sqrt{2}& \frac{3\sqrt{2}}{2}& \left(\sqrt{2},\frac{3\sqrt{2}}{2}\right)\\ \frac{\pi }{2}& 0& 3& \left(0,3\right)\\ \frac{3\pi }{4}& -\sqrt{2}& \frac{3\sqrt{2}}{2}& \left(-\sqrt{2},\frac{3\sqrt{2}}{2}\right)\\ \pi & -2& 0& \left(-2,0\right)\end{array}$

The orientation is 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 $0\le t\le 2\pi$.

As $t$ takes values from $0\le t\le \pi$, the corresponding orientation is from $\left(2,0\right)$ to $\left(-2,0\right)$.

Use points in the table to sketch the curve.

The graph of the parametric equation $x=2\cos t,y=3\sin t;0\le t\le \pi$ is

The orientation of the given curve is counter-clockwise.

Let $x=2\cos t,y=3\sin t$

$\Rightarrow \frac{x}{2}=\cos t,\frac{y}{3}=\sin t$

By using Pythagorean identity ${\sin }^{2}t+{\cos }^{2}t=1$,

$\Rightarrow {\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{3}\right)}^2=1$

$\Rightarrow \frac{x^2}{2^2}+\frac{y^2}{3^2}=1$

$\Rightarrow \frac{x^2}{4}+\frac{y^2}{9}=1$

The rectangular equation of the parametric equation $x=2\cos t,y=3\sin t;0\le t\le \pi$ is $\frac{x^2}{4}+\frac{y^2}{9}=1,y\ge 0$.