§ 4

Prove that the curve of intersection of a second-order surface with a plane that does not lie entirely on it is an algebraic curve of order at most two. How many common points can a line and a second-order surface have?

Find the equation and determine the type of the surface obtained by rotating about the axis of applicates the straight line:

Prove that the rectilinear generators of a hyperbolic paraboloid belonging to one family are all parallel to some plane.

On the hyperbolic paraboloid with the equation

there lie the parabolas $y=0, x^{2}=2 a^{2} z$ and $x=0, y^{2}=-2 b^{2} z$. Let the points $A_{1}$ and $B_{1}$ on the first parabola and the points $A_{2}$ and $B_{2}$ on the second all be at the same distance from the plane $z=0$.

Prove that the lines $A_{1} B_{2}, A_{1} A_{2}, B_{1} A_{2}$ and $B_{1} B_{2}$ are rectilinear generators.

Find the projection onto the plane $z=0$ of the curve of intersection of the two-sheeted hyperboloid $-x^{2}+y^{2}-z^{2}=1$ and the cone $5 x^{2}-3 y^{2}+4 z^{2}=0$.

Prove that no plane intersects an elliptic paraboloid in a hyperbola.