. Algebraic geometry; a new treatise on analytical conic sections . Tangents OP, OQ are drawn to the ellipse -2+?2 = l, centre C; prove that the product of the slopes of PQ and OC is - -r^- 12. Find the condition that the pole of lx + my=l with respect to theellipse ^+|j = l may Ue on the elUpse £^+^=1. 13. CP and CQ are at right angles, P and Q lying on an ellipse whosecentre is C: prove that __- + _^=_.+_^, where a and b are the semi- axes of the ellipse. [With the usual axes, let !=a, so that (CPcosa, CPsina) are theco-ordinates of P.] 14. If the chord of contact of tangents drawn to t
. Algebraic geometry; a new treatise on analytical conic sections . Tangents OP, OQ are drawn to the ellipse -2+?2 = l, centre C; prove that the product of the slopes of PQ and OC is - -r^- 12. Find the condition that the pole of lx + my=l with respect to theellipse ^+|j = l may Ue on the elUpse £^+^=1. 13. CP and CQ are at right angles, P and Q lying on an ellipse whosecentre is C: prove that __- + _^=_.+_^, where a and b are the semi- axes of the ellipse. [With the usual axes, let !=a, so that (CPcosa, CPsina) are theco-ordinates of P.] 14. If the chord of contact of tangents drawn to the ellipse -2+ia=lfrom the point (ajj, ^i) subtends a right angle at the centre, prove that 15, If O is the point {ocj, yi) and the straight line /=-. „=»• ^2 yi cos e sin e meets the ellipse — + j^=l at the points P and Q, prove that the reot- angle OP. 0Q=^:---t°K-1; ART. 208.] CONJUGATE DIAIMETEES. 195 208. To find-the locus of the middle points of a series of parallelchords of an ellipse. Let 0(x^,y.^) be the mid^point of any chord QQ of the Fig. 125. We may take for the equation of QQ cos 6 sin 6 x = Xj^ + rcosd, y = y^ + rsin6. n\2 4/2 .. where the chord meets the ellipse -2 + t2 = 1 we have, bysubstitution, ^ {x^+rviOsdY (y, + r sin Oy sin^^X „ /a;, cos 6 ViSin^N x,^ y-? , . Now O is the mid-point of QQ; .. the roots OQ, OQ, of thisequation are equal but of opposite sign; . a;^cos6 y^mvO_^ But (aij, j^i) is any point on the locus; .. suppressing sufBxes,iBcosfl vsin^ . ,,, is the equation of the locus, for ^ is a constant angle. 196 ■ THE ELLIPSE. [chap. x. This is a straight line through the centre, a diameter. (SeeDef. Art. 183.) If m is the slope of GQ, ta,nd = m, and the equation of CO52may be written y = —g— x. .. if y=mx bisects all chords parallel to y = mx, m = —5— or mm = —s-am a- Vice versd, if y = mx bisects all chords parallel to y = mx, wehave in the same way, mm = —^■ .. if the diameter POP bisects chords parallel
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