. Algebraic geometry; a new treatise on analytical conic sections . Fig. 169. Corollary 3. If the conic is a parabola, e=l and its equation is I -. n ^ - = 1 + cos ^, r 2ft „ ,^or with the usual notation, — = 2 cos jr, a ^ ftnaS - = COS d r 2 299. JVUh 8 as origin and SX as initial line, the equation, of the vrectrix is In - = e cos Let P(r, 6) be any point on the directrix. SX = SPcos^ = rcos 6, and SL = def.; 284 POLAR EQUATION OF A CONIC SECTION. [cHAt. xin. • ^ a .. - = r cos o, - = eoos ^ is the equation of the Fig. 170. Corollary. ^=n is the equation of
. Algebraic geometry; a new treatise on analytical conic sections . Fig. 169. Corollary 3. If the conic is a parabola, e=l and its equation is I -. n ^ - = 1 + cos ^, r 2ft „ ,^or with the usual notation, — = 2 cos jr, a ^ ftnaS - = COS d r 2 299. JVUh 8 as origin and SX as initial line, the equation, of the vrectrix is In - = e cos Let P(r, 6) be any point on the directrix. SX = SPcos^ = rcos 6, and SL = def.; 284 POLAR EQUATION OF A CONIC SECTION. [cHAt. xin. • ^ a .. - = r cos o, - = eoos ^ is the equation of the Fig. 170. Corollary. ^=n is the equation of SL the upper portion ofthe latus rectum. 0=--or 6 = -^ is the equation of the other portion. 300. To find the polar equation of the tangent to the conic- = 1 + « cos 9 at the point whose vectorial angle is a. Let -=Acos^ + Bsin^ be the equation of the chord through the points (rj, « + /?), {r^, a-/8) on the conic and near to oneanother. The point (rj, a + /?) is on the conic, and on the chord; •• -=l+ecos(a + ^), and - = Acos(a + /3) +Bsin(a + |8); ART. 80J.] POLAR EQUATION OF A CONIC SECTION. 285 .. A cos (a + /?) + B sin (a + /8) = 1 + e cos (a + ;8) or (A-e)cos(a + ;8) + Bsin(a + yQ) = l (1) In the same way, (A-e)cos(a-y8) + Bsin(a-/8)=l (2) Subtracting, -2{A-e)sina aff/S + 2 B cos a skrp = 0 and (A-e)sina-Bcosa = 0 (3) Now let the points move up to one another and becomes from (1) or (2), (A-e)cosa + Bsina- 1=0 (4) Solving (3) and (4) [see Art. 23], A-e_ B _ 1 cos a sin a sin^ a + COS^ oA = e + cosa, B = sina. T
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