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 . = SM, .-. SP = PM. And .\SP,PO = MP, PO, each to each, and OS* = OM, .-. the angle OPS = the angle OPM, .-. OP is the tangent at P. (Prop. VI.) So OP is the tangent at P. The points of contact P and P will be upon the sameor opposite branches of the hyperbola according as SM andSM require to be produced in the same or in oppositedirections with respect to S, in order to intersect thehyperbola. Prop. XIV.
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 . = SM, .-. SP = PM. And .\SP,PO = MP, PO, each to each, and OS* = OM, .-. the angle OPS = the angle OPM, .-. OP is the tangent at P. (Prop. VI.) So OP is the tangent at P. The points of contact P and P will be upon the sameor opposite branches of the hyperbola according as SM andSM require to be produced in the same or in oppositedirections with respect to S, in order to intersect thehyperbola. Prop. XIV. If from a point 0 a pair of tangents, OP, OP be drawnto an hyperbola, then the angles which OP and OF subtendat either focns will be equal or supplementary according asthe points of contact are in the same or opposite branches ofthe hyperbola. Let the points P and P be on opposite branches of thehyperbola. Join PS, SP; SP, SP. Produce PS to M, making PM equal to PS. Also fromPS cut off a part PM equal to P S. Join OM, OM; OS, OS. Then since OP, PS = OP, PM, each to each, and the angle OPS = the angle OPM, (Prop. VI.) .-. OS = OM, and the angle 0 SP = the angle OMP. 86 CONIC So OS = 031,and the angle OSP = the angle OMP,.-. OM = OM. Again, v SM = SP - SP = AA\and SM = &P - ,SP = ^1^,.-. &¥ = # v OS, SM = OS, SM, eacli to each,and OM = OM,.. the angle OSM = the angle OSM,and the angle OMS = the ansrle OSM is the supplement of 0£P,and 01P/S is the supplement of OMP,.. OSM is the supplement of 0SP,and 0JJ/Pthe supplement of 0J/P = OSP,and OMP = 0S-P,.-. OSPis the supplement of OSP. CONIC SECTIONS. 8f Hence the angles which OP and OF subtend either at Sor S are supplementary. In a similar manner if P and P are on the same branch ofthe hyperbola, the angles subtended either at 8 or S may beshown to be equal. Prop. XV. 55. If the tangent at any point P of an hyperbola meet theconjugate axis in the point t, and Pn be drawn at right anglesto CB; then
Size: 1717px × 1455px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
