Next, we need to find a restriction for u and v to describe only the part of the sphere that lies above the cone. We start by examining how the sphere and the cone intersect. Since both surfaces are symmetric around the z$-$axis, the sphere

x$2+$ y$2+$ z$2=144$

intersects the cone

z $=$

x$2+$ y$2$

in a circle. To find the equation of this circle, we begin by substituting

z $=$

x$2+$ y$2$

for z in the sphere equation, to get

x$2+$ y$2+$

x$2+$ y$2$

$2$

$=2$x$2+$ y$2=144.$

This can be further simplified to

x$2+$ y$2=.$