52 



ON THE EQUATION OF CURVES 



the diameter of the hyperbola which is parallel to it, and that 

 portion of the tangent is bisected at the point of contact. 



(5.) Since the diameter of a hyperbola which passes through 

 the point of contact is conjugate to the diameter which is 

 parallel to the tangent (X.a), it follows from the properties (y) 

 that the area of the triangle contained by any tangent and the 

 asymptotes, is equal to the area of the parallelogram contained 

 by the system of conjugate semidiameters one of which is 

 parallel to the tangent and the other passes through the point 

 of contact. Hence the area of the triangle in question is 

 the same for every tangent, and equal to 



F" sin y (B2 — AC)-4, (XI. y). 



(f.) If straight lines be drawn from the point of contact P 

 parallel to the asymptotes, the area of the parallelogram CP 

 formed by these lines and the asymptotes will evidently be 

 half the area of the triangle formed by the tangent and the 

 asymptotes. Hence, denoting the parallels by x and y, we 

 have (§) 



xy sin ^ = I F^ sin y (B^ — AC),-* 

 where ff denotes the angle contained by the asymptotes. If^ 

 for the sake of brevity, we assume 



c* = i F'^ sin y (sin d)-^ (B^ — AC)-* (a), 



the last equation becomes 



xy = lc' ..(36), 



which is the equation of the hyperbola referred to its asymp- 

 totes as axes of co-ordinates. 



(f.) Let m' and m'^ denote the direction-indices of the 

 asymptotes, then by the theory of the straight line 



tan^= ' ^^^~^j^ • 



I +m m +{m +m ) cos y 



Now since m' and m" are the roots of the equation 



Am' + 2Bm+C=o, (iv. ^), 



we shall have A {m'+ m") -\- 2 B=o, Am' m" — C=Oy 



and A {m—m") = 2 / (B^ — AC) ; 



