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

 Nov 6, 2019
 #1
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The possible values of k are 5 + sqrt(7) and 5 - sqrt(7), and their sum is 10.

 Nov 6, 2019
 #2
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Find the sum of all possible values of the constant k  such that the graph of the parametric equations
\(\begin{align*} x &= 2+ 4\cos s,\\ y &= k-4\sin s, \end{align*}\)
intersects the graph of the parametric equations
\(\begin{align*} x&=1+\cos t,\\ y&=-3+\sin t \end{align*}\)
at only one point.

 

\(\text{Let the center of circle 1, $~ \mathbf{c_1} = \dbinom{2}{k} $} \\ \text{Let the radius of circle 1, $~ \mathbf{r_1} = 4 $} \\ \text{Let the center of circle 2, $~ \mathbf{c_2} = \dbinom{1}{-3} $} \\ \text{Let the radius of circle 2, $~ \mathbf{r_2} = 1 $} \\ \text{$\mathbf{\overline{c_1c_2}}$ is the distance between $c_1$ and $c_2$ } \)

 

There is only one point, if \(\mathbf{|\overline{c_1c_2}| = r_1+r_2 }\)

 

\(\begin{array}{|rclcl|} \hline |\overline{c_1c_2}| &=& r_1+r_2 \\ |\sqrt{(2-1)^2+(k-(-3))^2} | &=& 4+1 \\ |\sqrt{1+(k+3)^2} | &=& 5 \\ 1+(k+3)^2 &=& 25 \\ (k+3)^2 &=& 24 \\ k+3 &=& \pm\sqrt{24} \\ \mathbf{k } &=& \mathbf{-3 \pm\sqrt{24}} & \text{or}& k=-3\pm 2\sqrt{6} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline k_1+k_2 &=& -3 +\sqrt{24}-3 -\sqrt{24} \\ k_1+k_2 &=& -6 \\ \hline \end{array} \)

 

The sum of all possible values of the constant k is -6

 

laugh

 Nov 6, 2019

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