Answer
$z=1$
Work Step by Step
As we are given that $e^{-x^2-y^2}-z=0$
Since, we have the vector equation $r(x,y,z)=r_0+t \nabla f(r_0)$
The equation of the tangent line for $\nabla f(0,0,1)=\lt 0,0,-1 \gt$ is
$0(x-0)+0(y-0)-1(z-1)=0$
or, $-z+1=0$
or, $z=1$