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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

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Find the sum of all possible values of the constant  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 28, 2018

#1
+5172
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We need to use trickery here.

The first set of equations is a circle of radius 4 centered at (2,k)

The second set is a circle of radius 1 centered at (1,-3)

A grazing solution will be such that the distance between their centers will be equal to the sum of their radii.

$$\text{we'll use the distance squared}\\ (2-1)^2+(k+3)^2 = (1+4)^2 = 25\\ 1+k^2+6k+9=25\\ k^2+6k-15=0\\ k=\dfrac{-6\pm\sqrt{36+60}}{2}\\ k = -3\pm 2\sqrt{6}$$

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Nov 28, 2018

#1
+5172
+2

We need to use trickery here.

The first set of equations is a circle of radius 4 centered at (2,k)

The second set is a circle of radius 1 centered at (1,-3)

A grazing solution will be such that the distance between their centers will be equal to the sum of their radii.

$$\text{we'll use the distance squared}\\ (2-1)^2+(k+3)^2 = (1+4)^2 = 25\\ 1+k^2+6k+9=25\\ k^2+6k-15=0\\ k=\dfrac{-6\pm\sqrt{36+60}}{2}\\ k = -3\pm 2\sqrt{6}$$

Rom Nov 28, 2018