. The principles of projective geometry applied to the straight line and conic . 290 Principles of Projective Geometry These loci intersect ia C{), K{AA .DB) and A\ B. The point Kis the pole of p with respect to (3) and is not a point on the locus. A, B are common conjugates of .1 and B and are points on the locus. Fordifferent positions of the line p, the loci of A and B are the lines LM and L3I. C is a fixed point. It has the same polar with respect to the three polars of every point on its polar pass through C. Hence the polar of C ispart of the locus. The locus, therefore,
. The principles of projective geometry applied to the straight line and conic . 290 Principles of Projective Geometry These loci intersect ia C{), K{AA .DB) and A\ B. The point Kis the pole of p with respect to (3) and is not a point on the locus. A, B are common conjugates of .1 and B and are points on the locus. Fordifferent positions of the line p, the loci of A and B are the lines LM and L3I. C is a fixed point. It has the same polar with respect to the three polars of every point on its polar pass through C. Hence the polar of C ispart of the locus. The locus, therefore, consists of three straight lines, viz. LM, LMand the common polar of EXAMPLES. (1) Two fixed conies touch at A and B. LM is a chord of one touching theother at Q. Prove that the line joining Q to the intersection of AM \di BL passesthrough a fixed point. Let LM meet AB in Q and let be P. Then the conies are in self-perspective with Q as centre of perspective and its polar QP as axis. In thisperspective MA and BL correspond as do the tangents at A and B. Therefore PQmust pass through their point of intersection which is a fixed point. (2) Through a fixed point 0 is drawn a chord GPP of a given conic, and on itis taken a point Q such that the range OPQP has a constant anharmonic that the locus of Q is part of a conic having double contact with the givenconic, and that if Q be the point in which OP meets the other part of the conic,the product (OQPF) (OQPP) is unity. See Art. 118. 3. (3) A variable tangent to a parabola meets two fixed tangents in P, Q, and thediameters through P, Q meet the curve in L, M. Determine the envelope of LM
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
