192 



MB. BOBERT FINT,AY ON 



hence we see (II, 8) that each of the curves (1) and (2) touches 

 the two straight lines represented by the equation 



(D»— AF)y* + 2 (DE — B F) a;y + (E^— CF) x^ = o...(5), 



the equation of the chord of contact being 



D^^ + E^ + F = o (6), 



which is therefore the polar of the given point A taken in 

 relation to each of the given curves. Hence if two conic 

 sections be stipplementary in relation to a given point, the 

 jpolars of the point taken in relation to the two conic sections 

 are coincident in direction. 



(c) By adding together equations (3) and (4), we obtain 



(Dy + Ear + F)* = o; 



from which we see that the curves (3) and (4) have a double 

 contact, the straight line (6) being the chord of contact. 

 Hence, two conic sections which are supplementary in relation 

 to a given point have a double contact, the chord of contact 

 being the polar of the given point in relation to each of the 

 curves. 



{d) When the given curve is a hyperbola, having its centre 

 at the given point, we have D = o and E = o, so that 

 equations (3) and (4) become 



A/ + ZBxy + Cx*+ F = o (7), 



A^* + 2Bxy + Cx' — F = o (8). 



In this case the curves have been called conjugate hyper- 

 bolas; hence we see that any two conjugate hyperbolas are 

 supplementary in relation to their common centre. 



(e) From any point P {x, y) in the curve (3) let P Q. be 

 drawn perpendicular to the straight line (6), and let P R^ and 

 P R^ be perpendicular to the straight lines (5) ; then (V, c) we 

 shall have 



Dar + E^ + F = PQ . /(D* + E*), 



(D' — A F) 2/' + 2 (D E — B F) a; y + (E' -. C F) ar' 



= M. PR/. PR,, 



