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
asked 2021-02-08
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.
\(\displaystyle{x}=-{7}{\cos{{2}}}{t},{y}=-{7}{\sin{{2}}}{t},{0}\le{t}\le\pi\)

Answers (1)

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.
0

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