. Plane and solid analytic geometry; an elementary textbook. int of AC, I) will recede indefinitely. When B coin-cides with the middle point of AC, it has no conjugateharmonic point. When B moves from the centre towardA, B comes in from the left toward A, It is desirable for our purposes to express the relationbetween these points in terms of distances from A the definition, AB x CD = AD x BC. SubstitutingCD = AD - A C and BC= A C - AB, this becomes ABxAD-ABx AC = AD x AC- AD x AB, Ari 2AB x ADAC= AB + AD Let the student show that BD = —— =r^r* Connect BA + BC these results with harm
. Plane and solid analytic geometry; an elementary textbook. int of AC, I) will recede indefinitely. When B coin-cides with the middle point of AC, it has no conjugateharmonic point. When B moves from the centre towardA, B comes in from the left toward A, It is desirable for our purposes to express the relationbetween these points in terms of distances from A the definition, AB x CD = AD x BC. SubstitutingCD = AD - A C and BC= A C - AB, this becomes ABxAD-ABx AC = AD x AC- AD x AB, Ari 2AB x ADAC= AB + AD Let the student show that BD = —— =r^r* Connect BA + BC these results with harmonic progression in algebra by show-ing that AC is a harmonic mean between AB and Ch. XII, § 84] POLES AND POLARS 155 84. Polar of a point. — The polar of a point with respectto any conic is defined as the locus of points which divideharmonically secants through the fixed point. The methods of finding this locus are the same for allthe conies. In problem 31, Chapter VIII, the studenthas been asked to find it for the circle by the aid of polar. Fig. 85. coordinates. The same method might be employed here,but it is thought best to use a very similar one, intowhich, however, polar coordinates do not enter. We shall find the polar of the point Px with respect tothe ellipse ^ + ^y = ^ Transform to the point Px as origin by the aid of the equations X = x + xv y = y + yv 156 ANALYTIC GEOMETRY [Ch. XII, § 84 The equation of the ellipse becomes b2x2 + a2y2 + 2 J2^ + 2 a2*/^ + J2^2 + a2y2 - a252 = 0. Let any line, y = Ix, through Px cut the ellipse in thepoints P2 and P3. We wish to find the locus of a pointP on this line, so situated that Pv P2, P\ and P3 forma harmonic range. By the theorem of Art. 14, Pv M2,M, and Mz will also form an harmonic range, and hence FM=2P,M2xP,M, y=? 2xA 1 PXM2 + PXMZ x2 + x3 If we start the solution of the equation of the line,y = lx, with the equation of the ellipse, we have (52 + a2^2) x2 + 2 (b2xx + «2^i) a; + ^i2 + a2^2 - a2&2 = 0, Erom
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