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



Now, if the tangents are drawn from any point (x\ y) to the conic S, 

 we may make that point our new origin, and by this transformation we 

 know that c is changed into the point-function S', while A and (ah — Ji^), 

 being translation-invariants, remain unaltered by the transformation ; 

 hence, as a generalization of (76), we are able to enunciate the following 

 general 



Theorem.— If from any point (x', y% tangents are drawn to the 

 conic 



ax^ 4- '2^'hxy-\' hy^ -[■2gx + 2fy + c=:0 

 the area of the triangle formed by the two tangents with their chord of 

 contact is 



(ab-h^) S'- A' "^ ^ 



where A is the discriminant and S' the point-function of the conic. 



A variety of particular theorems may be deduced from this general 

 formula ; thus, if the curve is a parabola, the area in question is 



and, if, further, the point from which the tangents are drawn be the 

 origin, we have the theorem that, if the general equation of the second 

 degree represents a parabola, and two tangents be drawn from the origin 

 to the curve, the area of the triangle formed by the two tangents and 

 the chord of contact is 



c\/c 



\/ a ^ cj\/h 

 Again, the chord of contact being the polar of {x\ y') with respect 

 to the conic, has for its equatio^ 



(ax-^hy'-i-g) x^ (hx-^hy' -{-f) y + gx'+fy'+c = 0, 

 and, therefore, if p be the perpendicular let fall on this chord from 

 (x\ y'), we have easily 



5)2 ~ C78) 



^ ~ iax'-^hy'-\-gy-^(hx' + hy'-^f)^ ^ ^ 



But, if D = be the equation of the director-circle of the conic, and, 

 therefore, D' its point-function, we have from (60) 



(a^'+ hy' + gy-V (hx'-^ %'+/)^ = (a+h) S' - D'. 

 Hence (78) gives 



^'=(Mlfs^^' (^^>- 



It is now easy to find the length of the chord intercepted between the 

 points of contact of the tangents, for if A be the length sought, we 

 have 



