. Algebraic geometry; a new treatise on analytical conic sections . Pio. 136. whence ^ «sinS^ ART. 232.] PROPERTIES OF THE ELLIPSE. 213 .. tan RSX ■■ RX esing(~°^) ^(e-cosg) SX a{l-e^)sme _ab{e-cos 6)62 sin 61 _«e-acos0_CS-CN SN6 sine PN ^PN = L RSX, PSN are *. = a rt. L. 232. The above proposition may also be proved geometrically. Let the chord PQ meet thedirectrix at R. Produce PS toP and draw PM, QK perpen-dicular to the directrix. PM PR SQ~~QK~QR from the similar A PRM, QRK. .. SR bisects the exteriorangle QSP- Let the chord turn about thepo
. Algebraic geometry; a new treatise on analytical conic sections . Pio. 136. whence ^ «sinS^ ART. 232.] PROPERTIES OF THE ELLIPSE. 213 .. tan RSX ■■ RX esing(~°^) ^(e-cosg) SX a{l-e^)sme _ab{e-cos 6)62 sin 61 _«e-acos0_CS-CN SN6 sine PN ^PN = L RSX, PSN are *. = a rt. L. 232. The above proposition may also be proved geometrically. Let the chord PQ meet thedirectrix at R. Produce PS toP and draw PM, QK perpen-dicular to the directrix. PM PR SQ~~QK~QR from the similar A PRM, QRK. .. SR bisects the exteriorangle QSP- Let the chord turn about thepoint P, so that the point Qmoves up to P, and ultimatelycoincides with it. Throughout the process = ^RSP. But when Q, coincides with P, PR becomes a tangent, andiPSR = ^PSR. .. each of these angles is then a right angle. .. PR, which is then a tangent, subtends a right angleat thefocus Fig. 136. 214 PROPERTIES OF THE ELLIPSE. [chap. X. 233. In an ellipse, tangents at the ends of a focal chord intersect onthe corresponding directrix. This may be proved as in Art. 166 for the parabola, or asfollows: The co-ordinates of any point on the directrix, may be taken to be (~, y^y The equation of the polar of this point is = 1. or - + ^i = Lae ¥ This passes through the point (ae, 0) the corresponding focus,and this proves the proposition. 234. If from T, any point on the tangent at P, perpendiculars TR, TM, are drawn to the. focaldistance SP, and the directrix,S R = «. T M. (Adams Proposi-tion.) Let the tangent meet thedirectrix at F. Join SF. Draw PK perpendicular tothe directrix. = art. L. .. RT is parallel to SP, SP~FP~ PK from the similar A FMT,FKP.
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