First off, when a sphere is cut by a plane, is a circle always formed or does a ellipse get formed in some cases? If a circle is always formed, how do you prove it?
Next, how would you find the area of the circle formed when the sphere described by the equation $$x^{2}+y^{2}+z^{2}\; =\; 5$$
is cut by the plane $$x+y+z=1$$? I know how you would do it if it was just a vertical or horizontal plane, but I am unsure of how to proceed in this case. I think this system of two equation with three variables must be simplified into the equation of a circle but am unsure of how to do it.