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 . n the triangles SPZ, 8PZ because the sides aboutthe angles SPZ, SPZ are proportional, and the anglesPSZ, PS Z are equal, being right angles, and the anglesSZP, SZP are each less than a right angle, .*. the triangles SPZ, SPZ are similar. {Euclid, VI. 7). .-. the angle SPZ = SPZ. Prop. VII. The tangents at the extremities of a focal chord intersect inthe directrix. Let PSQ be a focal chord, and let the ta
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 . n the triangles SPZ, 8PZ because the sides aboutthe angles SPZ, SPZ are proportional, and the anglesPSZ, PS Z are equal, being right angles, and the anglesSZP, SZP are each less than a right angle, .*. the triangles SPZ, SPZ are similar. {Euclid, VI. 7). .-. the angle SPZ = SPZ. Prop. VII. The tangents at the extremities of a focal chord intersect inthe directrix. Let PSQ be a focal chord, and let the tangent P meetthe directrix in Z. Join SZ; then the angle ZSP is a right angle, {Prop. V.) And .. also the angle ZSQ is a right angle, . •. ZQ is the tangent at Q. {Prop. V. Cor. 1.) Or the tangents at the extremities of a focal chord intersectin the directrix. Prop. VIII. 51. If the tangent at P meet the transverse axis in T, andPN be the ordinate of the point P; then CT . ON = GA\ Draw PMM at right angles to the directrices meetingthem in M and M. Join SP, SP: then since PT bisects the angle SPS, {Prop. VI.) .-. ST : ST :: SP : SP, {Euclid, VI. 3.) PM : PM, XlV: XK 80 CONIC .•. ST- ST + ST:: XN- XN: XN + XXor 2 GT : 2CS::2CX: 2CN,or GT: CS:: CX: CN=CS. CX, = CA\ {Prop. II.) 52. Def. The line PG, drawn at right angles to the tangentPT, is called the Normal to the hyparbola at the point P. Prop. IX. If the normal to the hyperbola at the point P meet thetransverse axis in the point G, and PN be the ordinate ofthe point P, then NG : NC :: EG1 : AC2. Draw PMM at right angles to the directrices, meetingthem in M and M, and produce SP to W; then sincethe angle TPG is a right angle,.. the angle WPG = the complement of the angle SPT,and the angle SPG = the complement of the angle SPT; CONIC SECTIONS. 81 but the angle SPT= the angle SPT,.-. the angle WPG = the angle SPG,.-. PG bisects the angle SPIV,.. SG : SG :: #P : SP, (Euclid, VI. A.):: PM : Pil/,:: XX:
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
