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 . angles ateither focus. Join SP, SP; SP, SP; and produce SP, SP to Mand M, making PM equal to SP, and PM equal to #P. Join OM, OM; OS, OS. Then since OP, PS = OP, PM, each to each. and the angle OPS = the angle OPIf, (Pvp. VII. Cor.) .-. 0S= OM, and the angle OSP = the angle OMP. So 0£ = <9JP, and the angle OSP = the angle OMP, .-. 0Jlf= OM. Again, v flfJf = SP+ SP= A A, and SiP = SP + SP = A A, .
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 . angles ateither focus. Join SP, SP; SP, SP; and produce SP, SP to Mand M, making PM equal to SP, and PM equal to #P. Join OM, OM; OS, OS. Then since OP, PS = OP, PM, each to each. and the angle OPS = the angle OPIf, (Pvp. VII. Cor.) .-. 0S= OM, and the angle OSP = the angle OMP. So 0£ = <9JP, and the angle OSP = the angle OMP, .-. 0Jlf= OM. Again, v flfJf = SP+ SP= A A, and SiP = SP + SP = A A, . SM, And v OS, SM = OS, SM, each to each, and 0M= OM, .-. the angle OSM= the angle OSM, and the angle OMS = the angle OMS. 48 CONIC SECTIONS. But the angle OMS = the angle OSPrand the angle OMS = the angle OSP,.-. the angle OSP = the angle OSP,.. OP and OP subtend equal angles at either focus. Prop. XVIII. 35. If from an external point 0 a pair of tangents OQ,OQ be drawn to an ellipse, and CO be joined meeting thechord QQ in V, and the ellipse in P; then (1.) QQ will be bisected in V. (2.) The tangent at P will be parallel to QQ. (3.) CP will be a mean proportional between CV and Produce OQ, OQ to meet the major axis produced in Tand T. Draw the ordinates NQ, NQ, and produce them to meetthe circle in q and q. CONIC SECTIONS. 49 Then Tq and Tq will be tangents to the auxiliary circle.{Prop. X.) Let Tq and Tq be produced to meet in o; join Co meetingthe chord qq in v, and the circle in p. Now, since the corresponding ordinates of the ellipse andauxiliary circle are in the constant ratio of BC to AC, thethree lines ol,pm, vn drawn at right angles to A A will passthrough the points 0, P, V respectively. For, according as 0 is the point where ol meets TQ orT Q we shall have 10 : lo :: NQ : Nq, :: BC : AC; or 10 : lo :: N Q : Nq, :: BC : AC, .?. Oo is perpendicular to A A. So P^j arj-d Vv are perpendicular to ^4^4, .-. Oo, Pp, Vv are parallel. Hence (1.) QV : VQ :: qv : »2. Bu
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
