An elementary treatise on coordinate geometry of three dimensions . Fig. 38. ellipse lies on OZ if c>0 and on OZ if c z—k. The hyperbola Is real for all real values of k, and its centre passes in turnthrough every point on zz. When /; = () the hyperbola degenerates into the two lines ~ — L = 0, 0=0. The sections of the surface by the planes z = k, z=—k project on the. Fio. 39. plane XOY into conjugate hyperbolas whose asymptotes are x2 II2z = 0, -j— r2=0- The sections by planes parallel to YOZ, ZOX are parabolas. The surface is the hyperbolic paraboloid,and tig. 39 shews the form and positi
An elementary treatise on coordinate geometry of three dimensions . Fig. 38. ellipse lies on OZ if c>0 and on OZ if c z—k. The hyperbola Is real for all real values of k, and its centre passes in turnthrough every point on zz. When /; = () the hyperbola degenerates into the two lines ~ — L = 0, 0=0. The sections of the surface by the planes z = k, z=—k project on the. Fio. 39. plane XOY into conjugate hyperbolas whose asymptotes are x2 II2z = 0, -j— r2=0- The sections by planes parallel to YOZ, ZOX are parabolas. The surface is the hyperbolic paraboloid,and tig. 39 shews the form and position of the surface fora negative value of c. 80. Conjugate diametral planes. An equation of theform ax2+by*=2z represents a paraboloid. Any line in the plane XOY whichpasses through the origin meets the surface in two co-incident points, and hence the plane XOY is the tangentplane at the origin. The planes YOZ, ZOX bisect chordsparallel to OX and OY respectively. Each is thereforeparallel to the chords bisected by the other. Such pairs ofplanes are called conjugate diametral planes of the paraboloid 124 COORDINATE GEOMETRY [ 81. Diameters. If A is the point (a, {3, y), and theequations to a line through A are x — a. y — 8 z — y , the distances from A to the points of intersection of theline and the paraboloid are given by r2(al2+6m2; + 2r(+bj3m -
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
