Plane and solid analytic geometry; an elementary textbook . 2 2 loid. If b = #, -- -f- &- = 2 cz represents a paraboloid of a a revolution about the Fig. 23. 256 ANALYTIC GEOMETRY OF SPACE [Ch. V, § 33 Let the student discuss the form of the surface repre- -v>2 n\lil nted by the equation — — ^- = 2czJ i a2 b2 hyperbolic paraboloid. (See Fig. 22.) -v>2 n\lil sented by the equation — — ^- = 2cz. It is called anJ L a2 b2 PROBLEMS 1. Prove that in both the elliptic and hyperbolic parabo-loids the sections parallel to the X-Z-plane are equal parab-olas ; also that the sections parall
Plane and solid analytic geometry; an elementary textbook . 2 2 loid. If b = #, -- -f- &- = 2 cz represents a paraboloid of a a revolution about the Fig. 23. 256 ANALYTIC GEOMETRY OF SPACE [Ch. V, § 33 Let the student discuss the form of the surface repre- -v>2 n\lil nted by the equation — — ^- = 2czJ i a2 b2 hyperbolic paraboloid. (See Fig. 22.) -v>2 n\lil sented by the equation — — ^- = 2cz. It is called anJ L a2 b2 PROBLEMS 1. Prove that in both the elliptic and hyperbolic parabo-loids the sections parallel to the X-Z-plane are equal parab-olas ; also that the sections parallel to the F-Z-plane are equalparabolas. 2. Show from the results of problem 1 that a paraboloidmay be generated by the motion of a parabola, whose vertexmoves along a parabola lying in a plane, to which the planeof the moving parabola is perpendicular; the axes of the twoparabolas being parallel, and (a) in the elliptic paraboloid, theirconcavities turned in the same direction; (b) in the hyperbolicparaboloid, their concavities turned in opposite directions. 3. Show that an ellipsoid may be generated by the motionof a variable ellipse, whose
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