PROJECTILE PARABOLAS. 25 



V^sin^a V'COS^a V^COS2a 



2g 2g 2g 



or F = — Dcosaa 



Now, if we eliminate a between 



y = — Dcos2a and x = Dsin2a 



the latter being the expression for the abscissa of the focus as 

 well as the vertex, since the axis of the parabola is vertical, 

 we obtain the equation of the locus of foci. 

 Squaring and adding 



x^' + y^ = D^ 



which is the equation of a circle, radius D, centre at origin. 



V. The loci of latus rectum extremities are given by 



y = F = — DcoS2a and x = Dsin2a dz 2Dcos"a 



the abscissae of the two ends of the latus rectum being found 

 by adding one-half of its length to, and by subtracting the 

 same quantit}' from, the abscissa common to vertex and 

 focus. We can successively write 



X H= 2Dcos^a — - Dsin2a 



X =P D(l + COS2a) := Dsin2a 



X ± y =P D = Dsin2a 



which, with y = — Dcos2a, on squaring and adding, gives 



(x ± y + D)'-' -f- y^ = D^ 



By transferring the origin to (=bD, o) and rotating the 

 axes through ^tan-i(=F2), or about =1=31° 43', this becomes 



(3 + V5)D^ (3-V5)D' 



