Im trying to compute the envelope of the family of planes $2a_1x+2a_2y-z+a_1^2+a_2^2$. So far, I got $\frac{dF}{da_1}=2x+2a_1=0$, $\frac{dF}{da_2}=2y+2a_2=0$, therefore $a_1=-x$, $a_2=-y$. Thus, the envelope is $z= -x^2-y^2$
Now I'm asked to draw a picture of this to illustrate the geometric meaning. But I can't even imagine how this would be in 3D. Can anybody help me out?
In cylindrical coordinates $x^2+y^2=r^2$, so you have $z=-r^2$. It's a paraboloid (you rotate the parabola around the $z$ axis. See this image