# 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

Question
Equations
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.
$$\displaystyle{x}=-{7}{\cos{{2}}}{t},{y}=-{7}{\sin{{2}}}{t},{0}\le{t}\le\pi$$

2021-02-09
Step 1
Given:
The following parametric equations, $$\displaystyle{x}=−{7}{\cos{{\left({2}{t}\right)}}},{y}=−{7}{\sin{{\left({2}{t}\right)}}},{0}\le{t}\le\pi$$.
Step 2
a) To eliminate the parameter to obtain an equation in x and y.
We have,
The following parametric equations,
$$\displaystyle{x}=−{7}{\cos{{\left({2}{t}\right)}}}$$....(1)
$$\displaystyle{y}=−{7}{\sin{{\left({2}{t}\right)}}}$$....(2)
Adding equations (1) and (2), we get
$$\displaystyle{x}^{{2}}+{y}{2}={\left(-{7}{\cos{{\left({2}{t}\right)}}}\right)}^{{2}}+{\left(-{7}{\sin{{\left({2}{t}\right)}}}\right)}^{{2}}$$
$$\displaystyle={49}{{\cos}^{{2}}{\left({2}{t}\right)}}+{49}{{\sin}^{{2}}{\left({2}{t}\right)}}$$
$$\displaystyle={49}{\left({{\cos}^{{2}}{\left({2}{t}\right)}}+{{\sin}^{{2}}{\left({2}{t}\right)}}\right)}$$
$$\displaystyle={49}{\left({1}\right)}\ldots.{\left.'{B}{y}{u}{\sin{{g}}}{t}{r}{i}{g}{o}{n}{o}{m}{e}{t}{r}{i}{c}{i}{d}{e}{n}{t}{\quad\text{if}\quad}{y}\right\rbrace}$$
=49
An equation in x and y is $$\displaystyle{x}^{{2}}+{y}^{{2}}={49}$$.
Step 3
b) To describe the curve and indicate the positive orientation.
We have,
$$\displaystyle{x}^{{2}}+{y}^{{2}}={\left({7}\right)}^{{2}}$$.
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,
$$\displaystyle{x}=−{7}{\cos{{\left({2}{t}\right)}}},{y}=−{7}{\sin{{\left({2}{t}\right)}}},{0}\le{t}\le\pi$$.
The interval given is $$\displaystyle{0}\le{t}\le\pi$$.
Therefore, for t=0,
$$\displaystyle{x}=-{7}{\cos{{\left({2}{\left({0}\right)}\right)}}}=-{7}{\left({\cos{{\left({0}\right)}}}\right)}=-{7}{\left({1}\right)}=-{7}$$
$$\displaystyle{y}=-{7}{\sin{{\left({2}{\left({0}\right)}\right)}}}=-{7}{\sin{{\left({0}\right)}}}{)}=-{7}{\left({0}\right)}={0}$$.
Implies the initial point is (−7,0).
$$\displaystyle{F}{\quad\text{or}\quad}{t}=\pi$$,
$$\displaystyle{x}=-{7}{\cos{{\left({2}\pi\right)}}}=-{7}{\left({1}\right)}=-{7}$$
$$\displaystyle{y}=-{7}{\sin{{\left({2}\pi\right)}}}=-{7}{\left({0}\right)}={0}$$
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.

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