a A2- 



1 h AB - cos o> 6 B2 - 1 



a + h — 21i cos 



0) A2 + B2 - 2 AB cos (0 - 2 sin^w 



c 



C2 





(e* - 2) sin^o) 

 - 02 ' 



ah-h^ 



sin^o) - ( A2 + BS - 2 AB cos w) 



c2 ~ 



c* 





(1-eS) sin^o) 



1887.] A. Mukhopadliyay — Memoir on Plane Analytic Geometry. 317 



we get 



(A2-1) a;2 + 2 (AB-cos co) a?i/4-(B3-l) 2/H2AC;j7 + 2BC7/ + C2 = 0, 

 a comparison of which with the standard equation 



ax^-\-'2hxy-{-hy'^ + 2gx-\-2fy + c = 

 gives 



whence 



and 



C4 



by substitution from the value of e^ in (73). These lead to the familiar 

 result 



(eg- 2)g _ (^+?>-2;^cos a))g 



§. 21. Area of a triangle. 



§. 21. Triangle formed by two tangents.— We now proceed 



to investigate the area of the triangle formed by two tangents drawn 

 from any point to the general conic, and the chord of contact. For this 

 purpose, we will first confine our attention to the simple case when the 

 tangents are drawn from the origin, and then an easy application of in- 

 variants will smoothly lead to the solution of the general problem. 

 The tangents which can be drawn from the origin to the conic 

 S = aa32 + 27?a?ij + %2 + 2^03 + 2/2/4-0=0 

 are given by (Salmon's Conies, §. 147, Ed. 1879, p. 149) 



(ac-/) 032 + 2 {cli^gf) xy+(bc-f) 2/2 = 0, ... (74) 

 and the chord of contact being the polar of the origin is 



gx+fy + c=:0. (75) 



The area of the triangle formed by the intersection of the lines in 

 (74) and (75) is at once written down by substitution in (31), viz., 

 (K ^2 - ^^ {af + lf-\-c-h^-2fgh - ahc) 

 ^^""^^ ^ " " ap -2fgh + bg^ ' 



which may be written 



c \/ —c A 



Area=-p r- (76) 



j (ab -h^) c- A [ 



