An elementary treatise on coordinate geometry of three dimensions . G 98 COORDINATE GEOMETRY [oh. vi. 19. Prove that the common generators of the cones(62c2 - a4) x2 + (c%2 - 64) 7/2 + (a262 - c4) ?2=0,be —a1 ca — b2ab — c2_ ax by cz lie in the planes {be ± a2)x+(ca ± 6% 4- (ab ±e2)z=0. 20. Prove that the equation to the cone through the coordinateaxes and the lines in which the plane lx+my + nz—0 cuts the coneax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy=0 is l(bn2+cm2 - 2fmn)yz + m(cl2 + an2 - 2^^)s^ + n(am2 + bl2 — 2hlm)xy=0. 21. Prove that the equation \ffx + \fgy + \fhz — 0 represents a conet
An elementary treatise on coordinate geometry of three dimensions . G 98 COORDINATE GEOMETRY [oh. vi. 19. Prove that the common generators of the cones(62c2 - a4) x2 + (c%2 - 64) 7/2 + (a262 - c4) ?2=0,be —a1 ca — b2ab — c2_ ax by cz lie in the planes {be ± a2)x+(ca ± 6% 4- (ab ±e2)z=0. 20. Prove that the equation to the cone through the coordinateaxes and the lines in which the plane lx+my + nz—0 cuts the coneax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy=0 is l(bn2+cm2 - 2fmn)yz + m(cl2 + an2 - 2^^)s^ + n(am2 + bl2 — 2hlm)xy=0. 21. Prove that the equation \ffx + \fgy + \fhz — 0 represents a conethat touches the coordinate planes, and that the equation to thereciprocal cone is fyz+gzx+hxy=0. 22. Prove that the equation to the planes through the originperpendicular to the lines of section of the plane lx+my + nz=0 andthe cone ax2 + by2 + cz2 = 0 is x2(bn2 + cm2)+y2{cl2 + an2) + z2(am2 + bl2) - 2amnyz - 2bnlzx — 2clmxy = 0. 804] CHAPTER VII. THE CENTRAL CONICOIDS. 64. The locus of the equation (1) x2 y2 z2 - (2) 9 9 9 a;2 y* z2 (3) cr b- c-. Pio Z . 29 We have shewn in §9 that the equation (1) representsthe surface generated hy the variable ellipse /,- 100 COORDINATE GEOMETEY [CH. VII. whose centre moves along ZOZ, and passes in turn throughevery point between (0, 0, — c) and (0, 0, + c). The surfaceis the ellipsoid, and is represented in fig. 29. The sectionby any plane parallel to a coordinate plane is an ellipse. Similarly, we might shew that the surface representedby equation (2) is generated by a variable ellipse whose centre moves on ZOZ, passing in turn throughevery point on it. The surface is the hyperboloid of onesheet, and is represented in fig. 30. The section by anyplane parallel to one of the coordinate planes YOZ or ZOXis a hyperbola.
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