. Algebraic geometry; a new treatise on analytical conic sections . I The equation of the tangent TQ is - = cos{6 - 6^ + ^) + eeos6. ART. S05.] POLAR EQUATION OF A CONIC this line passes through the pofeit {r^, 6^);■■ — = cos ;8 + e cos flj. Whence sec/3 = ■ e cos 6, .. substituting this value of sec;8 in (1), the required equation is - = e cos 6 + y—i li, which may be written ecosO, {--eooBdjl—ecos6A = oos{6-0i). 305. Tangents to a conicat the ends of a focal chordintersect on the directrix. If PSQ is the focal chord,and a the vectorial angle ofP, the vectorial angle of Qis TT + a
. Algebraic geometry; a new treatise on analytical conic sections . I The equation of the tangent TQ is - = cos{6 - 6^ + ^) + eeos6. ART. S05.] POLAR EQUATION OF A CONIC this line passes through the pofeit {r^, 6^);■■ — = cos ;8 + e cos flj. Whence sec/3 = ■ e cos 6, .. substituting this value of sec;8 in (1), the required equation is - = e cos 6 + y—i li, which may be written ecosO, {--eooBdjl—ecos6A = oos{6-0i). 305. Tangents to a conicat the ends of a focal chordintersect on the directrix. If PSQ is the focal chord,and a the vectorial angle ofP, the vectorial angle of Qis TT + a. .. the equations of the tangents TP and TQ are I r l_ r The second equation maybe written - = - cos(6 -a) + e cos 6. (2) - = COs(^-a) + 6COS0, (1); = cos(^ - JT - a) + ecos Flo. 173. At the point T wherethe tangents meet, we have from (1) and (2) by addition, - = e cos , But this is the equation of the directrix. .. the point T lies on the directrix, T 290 POLAR EQUATION OF A CONIC SECTION, [chap. xiii. Corollary. In a parabola, tangents at the ends of a focal chatd,intersect at right angles on the = 1 in the the equations of the tangents may be written = cos(5 - a) + cos 6 and -= -cos(5- a) + cos 6 r r I „ / n o\ C 1 Z r = 2cos(6-^jcos^ and -= -2sin(6-^jsin^. Now sin(5-0 = cosg + ^-g; .. the tangents are at right angles. (Art. 82.) 306. In a conic, the semi-latus rectum is a harmonic mean betweenthe segments of any focal chord. Let ~ = 1 + « cos 6 be the equation of the conic, and let PSQ be any focal chord. If a is the vectorial angle of the point P, tt + a is the vectorialangle of Q. The points P and Q are on the conic, .. —=l+«cosa, and -—= 1 +ecos(7r4-a) = l -recosa; •■• byaddition,- + —=2, i.
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