An elementary treatise on coordinate geometry of three dimensions . (x. + a2=0. §§ 100-1001 CONICOID THROUGB THREE GIT i.\ I. 107. Conicoids through three given lines. Th<equation to a conicoid, ax*+ty/i+cz2+2fyz + 2gzx + 2haay + 2ux+2vy+2 wz +(2=0, contains nine constants, viz., the ratios of any Dine of tin-ten coefficients a, b, c, ... to the tenth. Hence, since theseare determined by nine equations involving them, a coni-coid can be found to pass through nine given points. Butwe have proved that if three points of a straight line lieon a given conicoid, the line is a generator of t
An elementary treatise on coordinate geometry of three dimensions . (x. + a2=0. §§ 100-1001 CONICOID THROUGB THREE GIT i.\ I. 107. Conicoids through three given lines. Th<equation to a conicoid, ax*+ty/i+cz2+2fyz + 2gzx + 2haay + 2ux+2vy+2 wz +(2=0, contains nine constants, viz., the ratios of any Dine of tin-ten coefficients a, b, c, ... to the tenth. Hence, since theseare determined by nine equations involving them, a coni-coid can be found to pass through nine given points. Butwe have proved that if three points of a straight line lieon a given conicoid, the line is a generator of tin- a conicoid can be found to pass through anythree given non-intersecting lines. 108. The general equation to a conicoid through the twogiven lines u = 0 = v, u = 0 = v\ is \uu + juluv + wu + pvv = 0,since this equation is satisfied when u = 0 and v = 0, orwhen tv=0 and v — O, and contains three disposableconstants, viz. the ratios of A, /u., v to p. 109. To find the equation to the conicoid through tit n <given non-intersecting If the three lines are not parallel to the same plane,planes drawn through each line parallel to the other twoform a parallelepiped, (fig. 45). If the centre of the 164 COORDINATE GEOMETRY [ch. ix. parallelepiped is taken as origin, and the axes are parallelto the edges, the equations to the given lines are of theform, (1) y = b, z = — c; (2) z = c, x— — a; (3) x = a, y = — b,where 2a, 26, 2c are the edges. The general equation to aconicoid through the lines (1) and (2) is(y-b)(z-c) + \(y-b)(x+a) +iu(z+c)(z-c)+v(z+c)(x+a) = x = a, y= —b meets the surface we have/ulz2+2z(av-b)- //c2 + 2c (av + b) - 4<ab\ = 0,and if x = a, y = — b is a generator, this equation must besatisfied for all values of z. Therefore = 0 =- \ _c(av+b)_c a 2ab a and the equation to the surface is ayz + bzx + cxy+abc =
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