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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 $t$ goesfrom 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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hj

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.

hj

Hasan J.

April 4, 2020

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

The problem is this crab is motion of a particle with position acts Why, as team worries in keeping into a wall First, though we have axe minus off Squire Asli minus three square, it's the one true four. First we can scan each circle. It's a laugh on DH three and some readers of that curve. There's the code to two then not decided. One tea is between one and two. High tea is between high on too high. So we're half Why? This is between one hundred three. I won t is equal to one. Why he could too three axe equal to three. So here one tea is you go to two. Why couldn't you three on the axe? It could to selling So here face lies between one and three. So behalf asked him worries The particle is from this point to this point

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#### Topics

Parametric Equations

Polar Coordinates

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp