Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. x = -7 cos 2t, y = -7 sin 2t, 0 <= t <= pi

Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
$x=-7\mathrm{cos}2t,y=-7\mathrm{sin}2t,0\le t\le \pi$
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curwyrm

Step 1
Given:
The following parametric equations, $x=-7\mathrm{cos}\left(2t\right),y=-7\mathrm{sin}\left(2t\right),0\le t\le \pi$.
Step 2
a) To eliminate the parameter to obtain an equation in x and y.
We have,
The following parametric equations,
$x=-7\mathrm{cos}\left(2t\right)$....(1)
$y=-7\mathrm{sin}\left(2t\right)$....(2)
Adding equations (1) and (2), we get
${x}^{2}+y2={\left(-7\mathrm{cos}\left(2t\right)\right)}^{2}+{\left(-7\mathrm{sin}\left(2t\right)\right)}^{2}$
$=49{\mathrm{cos}}^{2}\left(2t\right)+49{\mathrm{sin}}^{2}\left(2t\right)$
$=49\left({\mathrm{cos}}^{2}\left(2t\right)+{\mathrm{sin}}^{2}\left(2t\right)\right)$

=49
An equation in x and y is ${x}^{2}+{y}^{2}=49$.
Step 3
b) To describe the curve and indicate the positive orientation.
We have,
${x}^{2}+{y}^{2}={\left(7\right)}^{2}$.
The curve represents a circle at the origin, (0,0) and radius, r=7.
To indicate the positive orientation, we will use the following parametric equations,
$x=-7\mathrm{cos}\left(2t\right),y=-7\mathrm{sin}\left(2t\right),0\le t\le \pi$.
The interval given is $0\le t\le \pi$.
Therefore, for t=0,
$x=-7\mathrm{cos}\left(2\left(0\right)\right)=-7\left(\mathrm{cos}\left(0\right)\right)=-7\left(1\right)=-7$
$y=-7\mathrm{sin}\left(2\left(0\right)\right)=-7\mathrm{sin}\left(0\right)\right)=-7\left(0\right)=0$.
Implies the initial point is (−7,0).
$F\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}t=\pi$,
$x=-7\mathrm{cos}\left(2\pi \right)=-7\left(1\right)=-7$
$y=-7\mathrm{sin}\left(2\pi \right)=-7\left(0\right)=0$
Implies the end point is (−7,0).
The initial point and the end point are equal, that is, (−7,0).
Hence, the orientation is positive in anticlockwise direction.