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Per the title, does a constant $C$ exist such that the surface of the paraboloid $z=x^2+y^2+C$ is tangent to the surface of the cone $x^2+y^2=z^2$? How would I find this constant?

Thanks a lot!

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    @ro44 No problem, and my answer is essentially the same as what joriki suggested you to do, since those two surfaces are rotational-invariant with respect to the $z$-axis, meaning if you try to look at them from any perspective perpendicular to the $z$-axis, you can't tell the difference, hence draw a cross-section viewing from $y$-axis of both surfaces, so that you could visualize it as to find $C$ such that at somewhere $z = x^2 + C$ is tangent to $z = \pm x$, the rest is the same.2012-05-03

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Yes. $C=1/4$. You can see this by following up on @Joriki's suggestion and solving the equivalent two dimensional problem: $y=r^2+C=|r|$ $y'=2|r|=1$