Step 1 Given: The following parametric equations, \(x=\ -7\ \cos(2t),\ y=\ -7\ \sin(2t),\ 0\ \leq\ t\ \leq\ \pi.\)
Step 2 a) To eliminate the parameter to obtain an equation in x and y. We have, The following parametric equations, (1)\(x =\ −7 \cos(2t)\)
\(y =\ −7 \sin(2t)\)(2) Adding equations (1) and (2), we get \(x^{2}\ +\ y^{2}=(-7\ \cos(2t))^{2}\ +\ (-7\ \sin(2t))^{2}\)
\(=49\ \cos^{2}(2t)\ +\ 49\ \sin^{2}(2t)\)
\(=49(\cos^{2}(2t)\ +\ \sin^{2}(2t))\)
\(=49(1)\ \because \text{By using trignometric identity}\)
\(=49\) 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, \(x^{2}\ +\ y^{2} = (7)^{2}.\) The curve represents a circle at the origin, \((0,\ 0)\ \text{and radius,}\ r = 7.\) To indicate the positive orientation, we will use the following parametric equations, \(x =\ −7\ \cos(2t),\ y =\ −7\ \sin(2t),\ 0\ \leq\ t\ \leq\ \pi.\) The interval given is \(0\ \leq\ t\ \leq\ \pi.\)
\(x =\ −7\ \cos(2(0)) =\ −7(\cos(0)) =\ −7(1) =\ −7\)
\(y =\ −7\ \sin(2(0)) =\ −7(\sin(0)) =\ −7(0) = 0.\) Implies the initial point is \((−7,\ 0).\) For \(t = \pi,\)
\(x =\ −7\ \cos(2\ \pi) =\ −7(1) =\ −7y =\ −7\ \sin(2\ \pi) =\ −7(0) = 0.\) 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.
