Answer
Elliptic Paraboloid
Work Step by Step
$x^{2}+y^{2}-2x-6y-z=10$
$x^{2}-2x+y^{2}-6y-z=-10$
$x^{2}-2x+1+y^{2}-6y+9-z=-10+1+9$
$(x-1)^{2}+(y-3)^{2}-z=0$
$z=(x-1)^{2}+(y-3)^{2}$
This is the equation of an elliptic paraboloid with vertex at (1,3,0)
Traces in the plane z=k are circles parallel to the xy-plane.
Traces in the planes x=k and y=k are parabolas parallel to the yz-plane and xz-plane, respectively.
