322 A. Mukhopadhyay- — Memoir on Plane Analytic Geometry. [No. 3, 

 But, substituting iovyi, 2/2 fi^oni (89) and (90) in (91), we liave 



jcj tan 6> 4-^2 tan <^ 



Now, assume 



a2 + 6nan^^ = 

 a2 + &s tan2<^ = 



C0S2 ff 



a'^ fjfi 

 cos2 <^' 



so that 



\3 = 1 _ e2 sin^ ^, /x2 = 1 _ e2 sin^ <p. 



a cos ^ a cos 



and a?! = — r — , jpg = • 



A fi 



Substituting these values in (94), we arrive at the following symmetrical 



theorem, viz., if 0, <p, \p be the angles of inclination of any two tangents 



to a conic and of their chord of contact to a directrix, we have 



A~"^ sin 6-\-ur^ sin 



tan i// = -— 4- , 



A coa ^+/A cos </> 



where the eccentricity of the conic is given by 



^^^l-X2^1-^2 



sin2<9 sin2(/)' 



(See Educational Times, November 1885, my Ques. 8337). 



§. 23. Applications. — To verify the truth of this theorem, we 



proceed to some applications. In the parabola, e = 1, so that 



A = cos ^, /x = cos <^, 



which give 



2 tan xp = tan ^+ tan </>, 



a result which can be proved independently, and is often useful in the 



elementary theory of projectiles. The particular case of the circle is 



specially interesting. Here e = 0, and A = />t= 1, whence 



sin ^4- sin <^ , 6-\-<p 



tan w = = tan — pr-, 



^ cos ^ + cos<^ 2 ' 



and 2xp=6^t 



or if/ — 6 = (p — xj/. 



To see the geometric meaning of this analytic condition, observe that, 

 in the circle, the foci coincide with the centre, and the position of the axes 

 becomes essentially indeterminate, while the directrix is situated at an 

 infinite distance. Now draw any two tangents OA, OB to a circle, and 

 let OA, OB, BA intersect the line at infinity in the points 0, D, E ; 

 Z.OCD = 0, ZODC = -<P, ZBEC = if/, 1> being taken negative as it is 

 measured in a direction opposite to that in which 0, xj/ are measured ; 



