OS" THE SECOND DEGREE. 



51 



XI. 



We have seen (vi. a) that if AO be a line parallel to an 

 asymptote of a hyperbola, cutting the curve in P3, and the two 

 parallel chords Pi Pj and Qi Q2 in A and O respectively, we 

 shall have 



APi.APa: OQ,. OQ2 : : APgiOPs. 

 Now, when A O coincides with an asymptote, A P3 and 

 O P3 become infinite, and may evidently be considered as 

 equal. Hence we see that, if any chord Pi Pj of a hyperbola 

 be drawn parallel to a given straight line, and produced if 

 necessary to meet an asymptote in A, the rectangle contained 

 by the segments into which the chord is cut by the asymptote is 

 invariable. 



(a.) When A coincides with the centre C of the hyperbola, 

 the points Pi and P2 may be real or imaginary, but the rect- 

 angle C Pj . C P2 is real, and equal to 



—E"(m^ + 2 w cosy + 1) : (Aw^ + 2Bm + C), (iv), 

 where m is the direction-index of Pj Pj . Hence (viii. a) the 

 rectangle A Pi . A P2 is equal to the square of the semi- 

 diameter which is parallel to Pi P2 . 



(/3.) If Pi P2 be produced to meet the other asymptote in 

 A', we shall have (a). 



AP, . AP2= A'Pi. A'Pa, 

 since each of these rectangles is equal to the square of the 

 semidiameter parallel to Pi P2. From this equation it is 

 evident that APi = A' P2, and thus we see that if any 

 straight line be drawn cutting a hyperbola and its asymptotes, 

 the segments intercepted between the curve and its asymptotes 

 shall be equal. 



{y.) When the chord Pi Pa becomes a tangent, the points 

 Pi and Pa coalesce in a point of contact P, and APi . AP2 be- 

 comes equal to AP ^ . Hence (a) if any tangent be applied 

 to a hyperbola and produced to meet the asymptotes y the part 

 of the tangent intercepted between the asymptotes is equal to 



