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 . gain, since CU : CP : CP : or, {Prop. XVIII.) .-. (7Z7 : CF : CP : CV2, (PWftV/, VI. 20 Cor) .-. CU-CV CU : CP - -CV2 : CP2, or UV CU ?. : PV .PV : CP2. Hence £ 7^2 CD : : PV . PV : CP2, or QV2 : PF PV : CD2 CP2. Peoi\ XXII. 39. The area of any parallelogram formed by drawingtangents to an ellipse at the extremities of a pair of conjugate 54 CONIC SECTIONS. diameters is equal to the rectangle contained b
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 . gain, since CU : CP : CP : or, {Prop. XVIII.) .-. (7Z7 : CF : CP : CV2, (PWftV/, VI. 20 Cor) .-. CU-CV CU : CP - -CV2 : CP2, or UV CU ?. : PV .PV : CP2. Hence £ 7^2 CD : : PV . PV : CP2, or QV2 : PF PV : CD2 CP2. Peoi\ XXII. 39. The area of any parallelogram formed by drawingtangents to an ellipse at the extremities of a pair of conjugate 54 CONIC SECTIONS. diameters is equal to the rectangle contained by the axes ofthe ellipse. Let PGP, DCD be a pair of conjugate diameters, andlet a parallelogram be formed by drawing tangents at thepoints P, P, D, Let the tangent at P meet CA produced in T; join D T. Draw the ordinates PN, DP, DR; then since PT isparallel to CD, the parallelogram PD is double the triangleCTD, and therefore equal to the rectangle contained by CTand DR. Now DR : CN :: BG : A C, (Prop. XIX. Cor.).-. CT . DR : CT . CN -.: RC : AC, (Euclid, VI. I.) :: BC . AC : AC*. (Euclid, VI. 1.)But CT . CN = A C\.-. CT . DR = AG . RG,*°. the parallelogram LL = 4 the parallelogram PD, = 4AC. RG,= AA . RR\ CONIC SECTIONS. 55 Cor. If PF be drawn at right angles to D CD meetingCD in F; then PF. CD = area of parallelogram PD,= Peop. XXIII. 40. If CP and CD be conjugate diameters, and PF bedrawn at right angles to CD meeting CA in 67, then PF. PG = BC2.
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
