172 



AXAI.YTir 



VIII. 



Fin. 70. 



;ilsn let the line XFX, pri p.-ndieular 



to ilie directrix, be the ar-a\i* : denote 

 the fixed distance ZF by 2f>, and let 



0, its middle point, IK- tin- origin of 

 coordinates; then tin- line OF, peivi 

 pendit -uhir to OX^ is the y-axis. Let 

 P= (#, y) bo any point on the cur\ < , 

 and draw LQP perpendicular to "). 

 also draw the onlinale MP, and the 



line FP. The line FP is called the focal radius of P. 



Then ZO=OF=p, 



and the equation of the directrix is x + /> = 0, . . 

 while the/0<?u is the point (p, 0). . . . 



Again, from the definition of the parabola, 

 -LP; [geometric equation] 

 * + *, and LP 



(1) 

 (2) 



ZO+ 



but FP=V( 



hence 



whence tf 2 = 4j>x, . . . [41] 



which is the desired equation. 



This first standard form [41] is the simplest equation of 

 the parabola, and the one which will be most used in tin- 

 subsequent study of the curve. It will be seen later 

 (Chapter XII) that any equation which represents a parab- 

 ola can be reduced to this form. 



104. To trace the parabola y~^ Jy>./. From equation 

 [41] it follows: 



(1) That the parabola passes through the point 

 way from the directrix to the focus. This point is 

 the vertex of the curve. 



(2) That the parabola is symmetrical with regard to tho 



