J M) . 1 \ A L YTIC OKOMETR T [On .XL 



Applying this transformation, equation (3) becomes 



aft + ggy+y* :r*-2sV + ./2 . 

 a 8 + P ci- -i /;- 



that is, dropping the accents, 



. . . 



which is the desired equation of tliu hyperbola wlim 



asymptotes as coordinate axes. 

 The equation of the conjugate hyperbola is then 





Remembering the relation 6 s = a a (e* 1), it will be .- 

 that tin- value of the constant term in equation (2) may be 



written 



. a* + V aV , 

 -4- - 



so tliat c is half the distance of the focus from the center of 

 the curve. Again, the coordinates of the foci, x = ae, y = 0, 

 become after the trans format ion (4), 



and the equations of the directrices, x = -, become 



(7) 



169. The tangent to the hyperbola a?y = c*. The equation 

 of the tangent to the hyperbola 



at any given point O,, y,), may be easily 1 ii\ d by the 

 secant method (cf. Arts. 84, 122). Let P } = (* y,) 

 P t =(z T y a ) be two points on the curve; then 



*,y, = c . . (2) and x# t = <?. . . (3) 



