$(\cos t,\sin t)$ is the constant speed parametrization of the unit circle. Adding to the coordinates shifts its center, while multiplying these terms scales the circle and hence changes its radius. So as you already recognized, $(1+\cos t,â3+\sin t)$ is a circle or radius $1$ with center $(1,-3)$. The other, $(2+4\cos s,kâ4\sin s)$, is a circle with center $(2,k)$ and radius $4$.
The fact that you have $-4\sin s$ instead of $+4\sin s$ means that the parametrization is the other way round from the conventional one: you start at the righternmost point, but then go down (negative $y$ direction) instead of up. Doesn't change which circles this describes, only what parameter $s$ corresponds to which point on that circle.
So you have two circles, and you want to know how to choose the $y$ coordinate for the center of the second in such a way that the two circles have exactly one point in common, i.e. they touch. There are two ways the circles might touch: they might touch on the outside, or the smaller one might touch the larger one from the inside. In the former case, the distance between the centers has to be the sum of the radii, in the latter case it's the difference of the radii. So you have touching circles for
$$ \sqrt{(2-1)^2+(k+3)^2}\in\{4+1,4-1\}=\{5,3\} $$
To solve this, square the equation and you get
\begin{align*}
(2-1)^2+(k+3)^2 &= 5^2 & (2-1)^2+(k+3)^2 &= 3^2 \\
k^2 + 6k + 9 &= 25 - 1 & k^2 + 6k + 9 &= 9-1 \\
k^2 + 6k - 15 &= 0 & k^2 + 6k + 1 &= 0 \\
k_{1,2} = \frac{-6\pm\sqrt{36+60}}{2} &= -3\pm2\sqrt6 &
k_{3,4} = \frac{-6\pm\sqrt{36-4}}{2} &= -3\pm2\sqrt2 &
\end{align*}
if you look at the parametric graph on the intervals $0 \le t \le \pi/4$ it does an arc staring at $(1,0)$ to $(-1,-1)$ on the next $\pi/4 \le t \le \pi/2$ it retraces that arc back to $(1,0)$ for $\pi/2 \le t \le 3\pi/4$ it does an arc that is the reflection of the old arc traversed for $0 \le t \le \pi/4 $ on the $x$-axis. and finally for $3\pi/4 \le t \le \pi$ it is back to $(1,0)$
so the period is $\pi.$
Best Answer
Formally, $4\pi$ is just the result of calculating the LCM of the periods of individual terms.
Recall that
The first term in both expressions repeats after $2\pi$ (i.e., it has a period of $2\pi$), and the second repeats after $\frac{4\pi}{11}$. So, after $4\pi$, both terms must repeat.