and 



Engler — The Normal to the Conic Section. 145 



as = +-^-n£ (19) 



a 2 — b 2 



*■-- ip=p* (20) 



Equation (19) represents a straight line parallel to the axis 



a? . a 2 



of ]Tand at a distance -a t- 2 £ from it. As -s r, is neces- 



er — o 1 a 1 — b l 



a? 

 sarily positive, the expression 2 , 2 £ always has the same 



sign as $ ; therefore, the asymptote represented by equation 



(19) always lies on the same side of the axis of Y as the 



a 2 

 point P. And as 2 , 2 is greater than unity, the point P 



lies between this asymptote and the axis of Y. 



The distance from the point P to this asymptote is 



„2 7,2 



Equation (20) represents a straight line parallel to the axis 



b 2 



of X and at a distance ^ — =v, 77 from it. 



a 2 — b 2 



b 2 . ., . 6 2 



As -3 — 72 is necessarily positive, the expression — —^ — j*v 



always has the sign opposite to that of t) ; therefore, the axis 

 of Xlies between the point P and the asymptote represented 

 by equation (20). 



The construction of the auxiliary hyperbola for this case is 

 similar to that already given for the parabola; but it will be 

 observed that neither of the asymptotes coincides with one of 

 the co-ordinate axes, and, therefore, a special construction to 

 find each of them is necessary. 



The asymptote parallel to the axis of Y will be found at a 



b 2 

 distance a _ , 2 $ beyond the point P (Fig. 5). To find it, 



join the end of the minor axis of the ellipse, B, with the 



