Problems

01

Hyperbolic paraboloids are ruled

Suppose that $(a,b,c)$ lies on the hyperbolic paraboloid $x^2-y^2+z=0$. Show that the lines given parametrically by the equations $(x,y,z)=(a+t,b+t,c+2(b-a)t)$ and $(x,y,z)=(a+t,b-t,c-2(b+a)t)$ both lie completely within the paraboloid. (Therefore this paraboloid is a ruled surface.)

02

Quadric translation to standard form

Convert the following two general equations of quadrics into standard form. Classify the surfaces they determine.

• (a) $x^2-y^2-z^2-4x-2z+3=0$
• (b) $4x^2+y^2+z^2-24x-8y+4z+55=0$