. Algebraic geometry; a new treatise on analytical conic sections . , iMPK = /.SPK and KP iscommon; .. ^PKM=aPKS. Similarly, from A MQK,SQK, = /lQKS. .. L PKQ = I (the fourangles at K) = a rightangle. MK .J R/^ <^ // X \ A /S ^ M ^ FlO. 99. 167. is a harmonic mean betweenthe segments of amy focalchord. Let PSQ be the focal chord and f—s, —) the Vm^ m/ co-ordinates of Fig. 100. The tangent at Q is at rt. l to that at P. ART. 168.] PROPERTIES OF THE PARABOLA. 151 .. the slope of the tangent at Q is — and the co-orduiatesof Q, are {am% - 2am). * SP=NX = a + -^, SQ= MX = a + (im2. t
. Algebraic geometry; a new treatise on analytical conic sections . , iMPK = /.SPK and KP iscommon; .. ^PKM=aPKS. Similarly, from A MQK,SQK, = /lQKS. .. L PKQ = I (the fourangles at K) = a rightangle. MK .J R/^ <^ // X \ A /S ^ M ^ FlO. 99. 167. is a harmonic mean betweenthe segments of amy focalchord. Let PSQ be the focal chord and f—s, —) the Vm^ m/ co-ordinates of Fig. 100. The tangent at Q is at rt. l to that at P. ART. 168.] PROPERTIES OF THE PARABOLA. 151 .. the slope of the tangent at Q is — and the co-orduiatesof Q, are {am% - 2am). * SP=NX = a + -^, SQ= MX = a + (im2. the harmonic mean between SP and SQ = + SQ = . n o ., = 2a = the semi-latus rectum. 168. Tangents at the ends of a chord of a parabola intersect on thediameter which bisects that chord. Let RP, RQ be the tangents, PQ the chord, V the middle pointof PQ. Let (iBj, y{) be the co- Jordinates of R. The equation of PQ, thechord of contact, isyi/^ = 2a{x + Zj). Where this meets thecurve y^ = 4aa;, we have bysubstitution, yvi = 2.(fc + ^x) or y^ - 2yy-^ + 4flKBi = 0. The roots of the quad-ratic are the ordinates ofP and Q. The ordinate of V, themiddle point of PQ = halfthe sum of the ordinatesof P and Q = half the sum of the roots of the quadratic = ^j = the ordinate of RV is parallel to the axis, which proves the theorem.
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Keywords: ., bookcentury1900, bookdecade1910, bookpublisherlondo, bookyear1916
