An elementary treatise on coordinate geometry of three dimensions . he ellipsoidwhose equation, referred to rectangular axes, is x2/a2+y2/b2+z2/c2=l,passes through the fixed point (0, 0, k). Shew that their line of inter-section lies on the cone x2(b2+c2-k2)+y2(c2+a2-k2) + (z-k)2(a2 + b2) = 0. Ex. 5. Tangent planes are drawn to the conicoid ax2 + by2 + cz2=lthrough the point (oc, 0, y). Prove that the perpendiculars to them /y>2 /\t& ^2 from the origin generate the cone ( + /3y + yz)2= — +4h—Prove that the reciprocal of this cone is the cone (ax2 + by2+cz2)(+b@2 + cy2-l)-(aoLX+b(3y +
An elementary treatise on coordinate geometry of three dimensions . he ellipsoidwhose equation, referred to rectangular axes, is x2/a2+y2/b2+z2/c2=l,passes through the fixed point (0, 0, k). Shew that their line of inter-section lies on the cone x2(b2+c2-k2)+y2(c2+a2-k2) + (z-k)2(a2 + b2) = 0. Ex. 5. Tangent planes are drawn to the conicoid ax2 + by2 + cz2=lthrough the point (oc, 0, y). Prove that the perpendiculars to them /y>2 /\t& ^2 from the origin generate the cone ( + /3y + yz)2= — +4h—Prove that the reciprocal of this cone is the cone (ax2 + by2+cz2)(+b@2 + cy2-l)-(aoLX+b(3y + cyz)2 = 0,and hence shew that the tangent planes envelope the cone (asc2+by2 + cz2 - l)(aoL2 + b(32 + cy2-l)-(aouc+b/3y + cyz-l)2=0. 69. The polar plane. We now proceed to define thepolar of a point with respect to a conicoid, and to find itsequation. Definition. If any secant, APQ, through a given point A,meets a conicoid in P and Q, then the locus of R, the har-monic conjugate of A with respect to P and Q, is the polarof A with respect to the Fig. 32. Let A, R (fig. 32) be the points (a, fi, y), (£ t], £), and letAPQ have direction-ratios I, m, n. Then the equations toAPQ are a; —q_y-/3_g —y I m n ?,70] THE POLAfi PLANE 106 and, measnree of ap and aq rXjt a i\alL-\-hin-~ -_ . - - - + i ?/- - - - - Let p be the measure of AR. Then, emee AP. AR AQ in harmonic progression, _ 2r^ _ _ . - - -- - -1 ?i + arjl + bSm + ] And from the equations to the line f-a = /p, rj — l8=mp, Hence the locus of i £ ; is the plane given byaaa — - —?/: = is called the polar plane of (oc jS - Cor. If A is on the surface, the polar plane of A is thetangent plane at A. The student cannot have failed to notice the similarity between theequations to corresponding loci in the plane and in space. There is aclose analogy between the equations to the line and the plane, thecircle and the sphere, the ellipse and the ellipsoid, the tangent orpolar and the tangent plane
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
