. Algebraic geometry; a new treatise on analytical conic sections . aosW 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 paralle
. Algebraic geometry; a new treatise on analytical conic sections . aosW 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 to the diameterDCD, then the diameter DCD bisects chords parallel to thediameter PCP*. Def. Conjugate diameters are such that each bisects chordsparallel to the other. 209. The twngents at the extremities of a diameter are parallel tothe chords bisect^ by that Fio. 126. Let the diameter CP bisect the chord QQ at V, and let thechord move parallel to itself, the point V approaching P. Thediameter always bisects the chord. Therefore when V coincideswith P, the equal portions QV, QV vanish together, and the chordbecomes the tangent at P. This proves the proposition. ART. 210.] CONJUGATE DIAMETERS. 197 210. Let POP, DCD be two conjugate diameters, and let 6and be the eccentric angles of the points P and D.
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