Engler — The Normal to the Conic Section. 147 



Through Q draw QR perpendicular to QP; then 



HR — QH • tan ^ = 7? tan^ ^ = ^_ ^ 7? ; ( 23 ) 



therefore, (eq. 20) the line through R parallel to the axis of 

 X is this asymptote. 



To find points on the curve the same construction can be 

 employed as in the case of the parabola, but the constant base 

 of the triaugles on the asymptote parallel to the axis of X is 



in this case —^ — j^^ and, therefore, equal to the distance -fi/P, 



in the figure. 



This hyperbola will intersect the given ellipse in two, three 

 or four points according to the position of the point P in the 

 plane. The actual construction of the hyperbola for any 

 given case will give all the real normals possible for the given 

 position of the point. 



When the point P is so situated that only one branch of 

 the hyperbola cuts the ellipse there will be two normals. 



When the point P is so situated that both branches of the 

 hyperbola cut the ellipse there will be four normals. 



When the point P is so situated that one branch of the 

 hyperbola cuts the ellipse while the other branch of the hy- 

 perbola is tangent to it there will be three normals. 



Both branches of the hyperbola cannot be tangent to the 

 ellipse, since one branch of the hyperbola passes through the 

 center of the ellipse, as is evident from equation (18); for 

 the same reason there must always be at least two real 

 normals, whatever the position of the point P. 



The following analysis shows in what way the number of 

 normals possible is dependent upon the position of the point 

 in the plane : — 



The equation of the tangent to the ellipse may be written 



y = _^x + -! (24) 



in which x', y' are the co-ordinates of the point of contact on 

 the ellipse. 



