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Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.

$ x = 5 + 2\cos \pi t $, $ \; y = 3 + 2 \sin \pi t $, $ \; 1 \leqslant t \leqslant 2 $

$x=5+2 \cos \pi t, y=3+2 \sin \pi t \quad \Rightarrow \quad \cos \pi t=\frac{x-5}{2}, \sin \pi t=\frac{y-3}{2}, \quad \cos ^{2}(\pi t)+\sin ^{2}(\pi t)=1 \Rightarrow$

$\left(\frac{x-5}{2}\right)^{2}+\left(\frac{y-3}{2}\right)^{2}=1 .$ The motion of the particle takes place on a circle centered at (5,3) with a radius $2 .$ As $t$ goes

from 1 to 2 , the particle starts at the point (3,3) and moves counterclockwise along the circle $\left(\frac{x-5}{2}\right)^{2}+\left(\frac{y-3}{2}\right)^{2}=1$ to

$(7,3) \text { [one-half of a circle }]$

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Hasan J.

April 4, 2020

–18 (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.

Hasan J.

April 4, 2020

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