What is the key difference in the equations of a hyperboloid of one sheet and a hyperboloid of two sheets?

The number of negative signs. A hyperboloid of one sheet has one negative sign, while a hyperboloid of two sheets has two negative signs.

Describe the quadric surface represented by the equation x^2 + y^2 = 4.

This equation represents a cylinder along the z-axis, where the cross-section is a circle of radius 2.

I can classify quadric surfaces from their equations.

1
The general equation for quadric surfaces includes terms up to the second degree and is represented as Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0.
2
An ellipsoid is defined by the equation x²/a² + y²/b² + z²/c² = 1, and when a=b=c, the ellipsoid is a sphere.
3
A hyperboloid of one sheet has one negative sign in its equation, while a hyperboloid of two sheets has two negative signs.