Question

The xy-plane intersects the sphere x^2+y^2+z^2+2x+2y-z=2 in a great circle.
True or false with full explanation

Answer 1

The xy-plane intersects the sphere $x^2+y^2+z^2+2x+2y-z=2$ in a great circle if and only if the center of this sphere belong to $xy$-plane. Let us rewrite the equation of the sphere in the the following form: $(x+1)^2+(y+1)^2+(z-\frac{1}{2})^2=2+1+1+\frac{1}{4}=\frac{17}{4}.$ It follows that $M(-1,-1,\frac{1}{2})$ is the center of the sphere. Taking into account that the third coordinate of $M$ is not equal to 0, we conclude that the center of the sphere does not belong to the $xy$-plane, and therefore, the sphere $x^2+y^2+z^2+2x+2y-z=2$ does not intersect the $xy$-plane in a great circle.

Answer: false