. The principles of projective geometry applied to the straight line and conic . envelope is required is P/and the envelope is therefore a conictouching the line at infinity, that is a parabola. IL Two points P and (^ are conjugate with regard to a conic. P lies on a fixedstraight line and PQ subtends a right angle at a fixed point. Prove that the locusof Q is a conic passing through the fixed point. Let PM be the fixed line on which P lies and 0 the fixed point. The polar QLof P will pass through a fixed point N, the pole of PM. The range described by Pis projective with the pencil described
. The principles of projective geometry applied to the straight line and conic . envelope is required is P/and the envelope is therefore a conictouching the line at infinity, that is a parabola. IL Two points P and (^ are conjugate with regard to a conic. P lies on a fixedstraight line and PQ subtends a right angle at a fixed point. Prove that the locusof Q is a conic passing through the fixed point. Let PM be the fixed line on which P lies and 0 the fixed point. The polar QLof P will pass through a fixed point N, the pole of PM. The range described by Pis projective with the pencil described by its polar XQ. Also the pencil OQ isprojective with the pencil OP and therefore with the range P. Hence the pencilsOQ and NQ are projective and the locus of ^ is a conic through 0 and X. 12. A, B,C are three fixed points. A circle is drawn through A, B and anotherthrough B, C. If these two circles have a fixed radical axis, prove that their respectivetangents at J, C meet on a fixed conic. H. p. G. 17 •258 Principles of Projective Geometry )f the circles. Tliroiigh A and. Let D be the second point of intersectii)nC draw AA\ CC parallel to BD. Thenthe pencils AD and CD for differentpositions of D are projective. But sinceangle AAD = BDA=BAP, the pencilsAD and AP are projective. Similarly the pencils CD and CP areprojective. Therefore the pencils Al*and CP are projective, and the locus ofP is a conic through A and C. 13. Given a straight line alongwhich the base of a triangle lies andan inscribed conic of the triangle, findthe locus of the vertex if the base alwayssubtends a right angle at a fixed point. Let ABC and ABC be two jiositions of the triangle. Since the base alwayssubtends a right angle at a fixed point the ends of the base are a pair of conjugatepoints of an involution determined by BC and EC. Hence the tangents from theends of the base always intersect on the straight line A A, which is therefore thelocus of the vertex. (Arts. 75 and 9) (c).) 14. Through a fixed
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