.i\.i/.rr/c QEOME'ii;) [Cm. \\. 



represents the polar of tin- point (a? r y^) with respect to the 

 hyperbola ; it represents the chord of contact if the point is 

 mi i side the hyperbola, and the tangent if the point is upon 

 the curve (Arts. 126, 122). Again, by the method slm\\n 

 in Art. 11:), merely replacing IP by 6*, it is evident that 



y = mr V 2 w a - V* . . . [i,7] 



is the equation of a tangent to the hyperbola in terras of its 

 slope m. The student will be able in like manner to p 

 other properties of the hyperbola, analogous to those already 

 shown for the ellipse, using the same methods of derivation. 

 It was shown, however, in the discussion of Chapter VIII. 

 as also in Art. I s . that the nature of the hyperbola appar- 

 ently differs widely from that of the ellipse, consisting, as 

 it does, of two open infinite branches instead of one cl< ><! 

 oval. It is desired in the present chapter to show some of 

 the most important properties of the hyperbola which corre- 

 spond to similar properties in the ellipse ; and also to p 

 some special properties which are peculiar to the hyperbola* 

 For the most part, these will be derived for the hyperbola 



^ = 1 ; and the facts summarized above will be assumed. 

 a* (r 



161. The difference between the focal distances of any point 

 on an hyperbola is constant ; it is equal to the transverse axis. 



The hyperbola -- %- = 1 has its foci at the points 

 a 3 (r 



=(- ae, 0), F 2 = (ae, 0), with ft 2 = aV 1 - a 2 . 



Let P 1 = (x r #j) be any given point on the curve, so that 



