174 ANALYTIC' GEOMI.lltY VIM. 



If from any point on the parabola, a perpendicular /> 

 to the axis of the curve, the square on tin's /><'r/><'n<tt<'n!<tr ?R 

 the rectangle forme </ /<// /// latus rectum <///,/ // 

 'rom tfo vertex to the foot of the perpendicular. 



This geometric property enables one to write down immedi- 

 ately the equation of the parabola, whenever the axis of 

 the curve is parallel to one of the (<><>r<lin ; , 



E.g.* if the vertex of the parabola is the point A = (A, &), 

 and its axis is parallel to the ar-axis, as in the figure, let 



F be the focus and P = (x, y) 

 be any point on the parabola; 

 draw MP perpendicular to the 

 axis AK. Then 



-ft),. [42] 



M AI "^ 



which is the equivalent al^eln-aie 



equation. This may be taken as 



a second standard form of the equation, representing tl it- 

 parabola with vertex at the point (A, &), with axis parallel 

 to the x-axis, and, if p is positive, lying wholly on the posi- 

 ti\v side of the line x = k 



Equation [42] evidently may be reduced to equation [1 1 ] 

 by a transformation of coordinates to parallel axes through 

 the vertex (A, A:), as the new origin. 



Again, suppose the position of the parabola to be that 



represented in Fig. 78. The vertex is A = (A, &), and the 



of the parabola is parallel to the y-axis. Let P = (x. y) 



l>e any point on the curve, and draw MP perpendicular to 



th axis of the curve. 



Then KP 2 = 4 AF - AM [geometric property] 



= 4 p AM, [here p is negai 



