I've had this problem for a while, but I have not been able to solve it. Any help would be really appreciated, thanks in advance!

Problem: Find the sum of all possible values of the constant $k$ such that the graph of the parametric equations \begin{align*}x &= 2+4\cos s,\\ y &= k-4\sin s,\end{align*} intesects the graph of the parametric equations \begin{align*}x&=1+\cos t,\\ y&=-3+\sin t\end{align*} at only one point.

  • $\begingroup$ Hint: Describe the graphs of the parametric equations. $\endgroup$ – Blue Sep 14 '14 at 22:24

The first curve is a circle with radius $4$ and centre $(2,k)$, the second curve is a circle with radius $1$ and centre $(1,-3)$. The solution are given by the points on the $x=2$ line such that the distance from $(1,-3)$ is three or five (i.e $4\pm 1$), so we have four solutions, and they are symmetric with respect to the $y=-3$ line, so the sum of all the possible $k$'s is just $-12$.

  • 1
    $\begingroup$ Very neat! I especially like the way you avoid actually finding the values of $k$ by using symmetry. By the way, I advise writing your last sentence with $k$'s rather than $k$s: the lack of apostrophe confused me briefly, when I thought about $k$ times s. $\endgroup$ – Rory Daulton Sep 14 '14 at 22:35

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