A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . ini£ and ^ ; then Since Br is bisected in Q, (ProjJ. XIX. Cor. 1.) .*. Pr is bisected in q, and PA = RK, {Prop. XIX. Cor. 2.) .. Kq is parallel to Pr, .. CT : CX:: Cr : <7<?, :: Ch : CV, .-. C7. CT = CK. Ch = CP\ Cor. 1. Conversely, if QVbe an ordinate to PV,and CV . CT = CP2,then Q T is the tangent at Q. Coe. 2. Hence also, if RE meet the curve in IT and £7,and h IT, h IP be drawn,since CK . Ch =
A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . ini£ and ^ ; then Since Br is bisected in Q, (ProjJ. XIX. Cor. 1.) .*. Pr is bisected in q, and PA = RK, {Prop. XIX. Cor. 2.) .. Kq is parallel to Pr, .. CT : CX:: Cr : <7<?, :: Ch : CV, .-. C7. CT = CK. Ch = CP\ Cor. 1. Conversely, if QVbe an ordinate to PV,and CV . CT = CP2,then Q T is the tangent at Q. Coe. 2. Hence also, if RE meet the curve in IT and £7,and h IT, h IP be drawn,since CK . Ch = CP2, .. k £7 and & £T are tangents to the hyperbola at IT and IP. Trot. XXVIII. 65. If two chords of a hyperbola intersect one another, therectangles contained by their segments are proportional tothe squares of the diameters parallel to them. Let Q Oq be any chord drawn through the point 0, andlet CD be drawn parallel to it, meeting the conjugatehyperbola in D. Produce Qq to meet the asymptotes in E and r; and drawthe diameter CP V, bisecting both Qq and Er in V. (PropXIX. Cor. 2.) Also draw the tangent LPl parallel to Qq, meeting theasymptotes in L and /. 102 CONIC Now since Qq is divided equally in V and unequally in 0,... QO . Oq = Q V2 - 0 V2; {Euclid, EL 5.)so also RO . Or = R V2 - 0 V2 ? {Euclid, II. 5.)... Oq = RV*-QV*, = RQ . Qr {Euclid, II. 5.)= Pi2, (Prop. XX.).-. = RO . Or -PL2. Again, through 0 and P draw POe J7PIF, at rightangles to the axis meeting the asymptotes in P, e, Z7, IF;then RO : OP : and rO : Oc : PC>. rO : OE . Oe : PP : P£T,PI : PTF,PP2 : PIT . PW; CONIC SECTIONS. 103 but PIT. PW = BC\ {Prop. XVI.)and PL* = CP2, (Prop. XXIII.) .-, BO . rO : OE . Oc :: CD2 : BC\ or : CD2 :: OE . Oe \ BG\ - PL* : CP2 :: OE . Oe - BG2 : BC\ or QO . Oq : CD2 :: OE . Oe - BG2 : BC\ In trie same manner if through 0 another chord Q Oqbe drawn, and CD be drawn parallel to it, meeting the con-jugate hyperbola in D, we shall have
Size: 1597px × 1565px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
