1 State whether the given equation is an ellipsoid, a hyperboloid, an elliptic cone or a paraboloid. If it is a hyperboloid, state whether it is of one sheet or two sheets.

x 2 + 2y 2 + 3z 2 = 1

2. Let three planes x = 1 2 , x = 1, and x = 2 be given. Find the equation of the trace of the above quadric surface in each of them, and state whether it is a point, a conic section, or nothing. If it is a conic section, state whether it is an ellipse, parabola, or hyperbola

Problem 1 1. State whether the given equation is an ellipsoid, a hyperboloid, an elliptic cone or a paraboloid. If it is a hyperboloid, state whether it is of one sheet or two sheets. $2+2y2+322=1 2. Let three planes :1: = %, r = 1, and .1'; = 2 be given. Find the equation of the trace of the above quadric surface in each of them, and state whether it is a point, a conic section, or nothing. If it is a conic section, state whether it is an ellipse, parabola, or hyperbola. Problem 2 1. The following two equations are hyperboloids. State whether each of them is of one sheet or two sheets. {a}:11722y2z2=1and{b]:z22y2r2=1 2. Determine whether a statement in the following two is true or not, and justify your answer with a picture or an equation. (1) If one rotates (a) by the angle g from the positive ar-axis to the positive z-axis, it becomes (b). (2) (a) has an ellipse as its trace in the plane :1: = 2