Answer
The intersection is a circle centered at the origin, radius $2$, in the plane $y=x$.
Work Step by Step
$x^{2}+y^{2}+z^{2}=4$
is a sphere centered at the origin, radius $2$.
$y=x$ is a plane perpendicular to the xy-plane (contains the z axis).
See image below.
The intersection is a circle centered at the origin, radius $2$, in the plane $y=x$.
