## Related questions with answers

Find parametric equations and a parameter interval for the motion of a particle starting at the point (2, 0) and tracing the top half of the circle

$x ^ { 2 } + y ^ { 2 } = 4$

four times.

Solutions

VerifiedAt the beginning of the task, we notice that the given equation is the equation of the circle with the center at the origin and the radius $r=2$.

Since the particle moves on the upper part of the circle, then we will draw only the part of the circle that is above the $x$-axis.

$\small \text{Figure $1$. The part of the circle above the $x$-axis}$

*How do we write a circle as a parametric equation?*

We can let $x = 2\cos t$ and $y = 2\sin t$ since $(2\cos t)^2 + (2\sin t)^2 = 4\cos^2 t + 4\sin^2 t = 4$, i.e., $x^2 + y^2 = 4$.

However, since we only need the top half of the circle, we want $y$ to be always positive, therefore, we can modify $y$ to be $2\left| \sin t\right|$.

Now, at $t = 0$, $(x, y) = (2, 0)$ and at $t = \pi$, $(x, y) = (-2, 0)$, that is, on the interval $[0, \pi]$, it traces the top half of the circle once counterclockwise. The second time will be on the interval $[\pi, 2\pi]$ when it traces the semicircle clockwise from $(-2, 0)$ at $t = \pi$ to $(2, 0)$ at $t = 2\pi$. The third time will be on the interval $[2\pi, 3\pi]$ when it traces the semicircle counterclockwise from $(2, 0)$ at $t = 2\pi$ to $(-2, 0)$ at $t = 3\pi$. Last but not least, the fourth time will be on the interval $[3\pi, 4\pi]$ when it traces the semicircle clockwise from $(-2, 0)$ at $t = 3\pi$ to $(2, 0)$ at $t = 4\pi$.

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