. Algebraic geometry; a new treatise on analytical conic sections . in this case. The proposition may be proved in the same way with anyother equations. 52. To find the equation of the straight lines joining the origin tothe points of intersection of ax^ + 2hxy + by^ + 2gx + 2fy + c = 0 (1) and lx + + n = 0 (2) The required equation will be homogeneous and of the seconddegree. In equation (1), multiply the terms of the second degree by (- n)\ „ „ first „ -{lx + my)n, „ constant term by (Ix + my). \ These three expressions are equal for lx + my= -n.] The result is n2(aa;2 + 2hxy + by^)
. Algebraic geometry; a new treatise on analytical conic sections . in this case. The proposition may be proved in the same way with anyother equations. 52. To find the equation of the straight lines joining the origin tothe points of intersection of ax^ + 2hxy + by^ + 2gx + 2fy + c = 0 (1) and lx + + n = 0 (2) The required equation will be homogeneous and of the seconddegree. In equation (1), multiply the terms of the second degree by (- n)\ „ „ first „ -{lx + my)n, „ constant term by (Ix + my). \ These three expressions are equal for lx + my= -n.] The result is n2(aa;2 + 2hxy + by^)- 2n{lx + my){gx +fy) + cQx + myf = 0. ...(3) ART. 53.] PAIRS OF STRAIGHT LINES, ETC. 67 Now any values of x and y which satisfy equations (1) and (2)satisfy (3) also, for it is formed by combining equations (1)and (2); .. the locus represented by (3) passes through the commonpoints of (1) and (2). Also equation (3) is homogeneous and of the second degree,and therefore represents two straight lines through the origin; ., it represents the lines PlO. 36. If PRQ is the curve represented by (1), PQ the straight linerepresented by (2), (3) is the equation of the straight linesOP, OQ. 53. A homogeneous equation of the n degree represents n straightlines, which all pass through the origin. When the coefficient of y, the highest power of y in theequation, is reduced to unity, the equation may be written y + ajy~^x + a^^V + ... + a„a; = 0. Dividing through by a, this becomes l-l Hfy +---+«„=o., •(1) Treating this as an equation for ^, it has n roots, for it is ofthe M. degree. * Let JWi, wij, wig, ... m„ be the roots. * 58 PAIRS OF STRAIGHT LINES, ETC. [chap. lil. Then the equation (1) must be identical with Hence if (x, y) is any point on any one of the straight lines,y-miX = 0, y-m^x = 0, ...y-m„x = 0, those values of x and ysatisfy equation (1), for one of the factors of (2) will be zero; .. the gi^en equation represents the n straight lines y - m^x = 0
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