Engler — The Normal to the Conic Section. 145 



x = -^-^,^, (19) 



a^ — b^ 



and 



y = - 77. (20) 



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



of Fand at a distance g a2 ^ from it. As o 52 '^ neces- 



a2 

 sarily positive, the expression 3 ^^ ^ always has the same 



sign as f ; therefore, the asymptote represented by equation 



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



(i^ • • -rt 



point P. And as -j — jo, 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 



d^ — 6^ a^ — 6' 



e. (21) 



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



of X and at a distance ^ — r? V from it. 



a^ — b^ 



As -2 72 is necessarily positive, the expression — ^ ,^ V 



always has the sign opposite to that of ?; ; 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 



62 

 distance ^ , 2 ^ beyond the point P (Fig. 5). To find it, 



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



