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.

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.