. Algebraic geometry; a new treatise on analytical conic sections . Fig. 61. Let P be the point (Xj, y^, and PAB any chord through P;AQ, BQ the tangents at A and B. It is required to find the locus of Q. Let {h, h) be the co-ordinates of Q. Then the equation of itschord of contact AB is (Art. 93) xh + yk^a^. 86 THE CIRCLE. But (Xp ^i) is on this line; But (h, k) is any point on the locus; , .. the equation of the locus is x^x + yj^y = a^or xx-^ + yy-^ = a^, a straight line. [chap. Fio. 62. JVken (a^, y^ is outside the circle, the polar is the same as thechard of contact of tangents dr
. Algebraic geometry; a new treatise on analytical conic sections . Fig. 61. Let P be the point (Xj, y^, and PAB any chord through P;AQ, BQ the tangents at A and B. It is required to find the locus of Q. Let {h, h) be the co-ordinates of Q. Then the equation of itschord of contact AB is (Art. 93) xh + yk^a^. 86 THE CIRCLE. But (Xp ^i) is on this line; But (h, k) is any point on the locus; , .. the equation of the locus is x^x + yj^y = a^or xx-^ + yy-^ = a^, a straight line. [chap. Fio. 62. JVken (a^, y^ is outside the circle, the polar is the same as thechard of contact of tangents dravm from (aij, y-^. For the geometrical construction of the polar of a given point,see Baker and Bournes Geometry, p. 371. 95. If the polwr of the point P passes through the point Q, the pola/rof the point Q, passes through P. Let (x^, y^) be the co-ordinates of P, (x^ y^ those of equation of the polar of Pwith respect to the circle is xx-^ + yy-^ = a^. ^RT. 98.] THE CIRCLE. 87 This passes through the point Q; .•. x^^^-y-^y^ = a^ (1) But aa;2 + 2/y2 = a2 is the polar of Q, arid by (1), this straightline passes through {x^^y^P, which proves the proposition. 96. Given that kx, + ^y + G = Q is a tmgent to the circle x^ + y^ = a\find the co-ordinates of its point of (a^, ^j) be the co-ordinates xx^ + yy^ = a^ is the equation of the tangent;.. this equation must be identical with kx + ^= -C,for the two equations represent the same straight line;..
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