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WZ
Numerade Educator

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Problem 21 Medium Difficulty

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

$ x = 5\sin t $, $ \; y = 2\cos t $, $ \; -\pi \leqslant t \leqslant 5\pi $

Answer

$x=5 \sin t, y=2 \cos t \Rightarrow \sin t=\frac{x}{5}, \cos t=\frac{y}{2} \cdot \sin ^{2} t+\cos ^{2} t=1 \Rightarrow\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{2}\right)^{2}=1 .$ The motion of the
particle takes place on an ellipse centered at $(0,0) .$ As $t$ goes from $-\pi$ to $5 \pi,$ the particle starts at the point (0,-2) and moves
clockwise around the ellipse 3 times.

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Video Transcript

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