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Suppose we have a sphere with radius $3$ centered at $(-3,3,2)$. What is the equation of the trace of this sphere on the $xy$ plane?

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Write down the equation of the sphere. Intersect it with $xy$ by substituting $z = 0$. You'll get a circle of center $(-3, 3, 0)$ and radius $\sqrt{5}$. The projection of the sphere is concentric but has bigger radius: $3$. – user2468 Aug 29 '12 at 15:53

Hint: The center projected onto the $xy$-plane is $(-3,3,0)$. So you get a circle of radius $3$ with that center.

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It is described by equations of a circle of radius $3$ and the center in $(-3,3,0)$ contained in the plane $XY$. So it is:

$$(x+3)^2+(y-3)^2=9\ \ \ \mbox{and}\ \ \ z=0.$$

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