226 ANALYIK- GXOXBTHY [('.,. 1\. 



< 1 The focus is equidistant from the points P^ T, and JV~. 

 Por PP, I.J\ /.! - .U/ 1 =/- 1 +/> f 



and J P^=^Af 1 -j-(-af 1 ^-yl/ t ; = - r i-H^; Art. 187 



hence FP l =TF=FX. 



The point F \* the midpoint of the hypotenuse of the 

 right triangle TP^ and is therefore equidistant from tin- 

 vertices T* ./Y and ^ T - Thus a tliird method is suggested for 

 constructing the tangent and normal at 7' r \i/.. : ly means r 

 a circle, with the focus JP as center, and the focal radius J'/\ 

 as radius, which cuts the axis in Tand N. 



(2) The tangent and normal bisect internally and externally \ 

 respectively, the angle between the focal radius to the pain 

 contact and the perpendicular from that point to t/< 



1 .r ZZ 1 P,r=ZP 1 7 r F, since L^ || TF\ 

 and /. TP^F = ^.P l TF, since TF = FP l ; 



Also, Z FP l tf=^ NP& since 

 (3) Through any point in the plane two tangents can be 

 drawn to the parabola (cf.'Arts. 89, 



The line y = mx+ (1) 



m 



is tangent to the parabola y 1 = 4 px for all values of m. If 

 P'= (z'i y') be any given point of the plane, then tin: tan- 

 gent (1) will pass through P' if, and only if, m satisfy 

 equation 



y' = 



,, if m ,y y ^-12L. . (2) 



Therefore two, and only two, values .f m satisfy the given 

 conditions; and therefore through any point of the plane two 



