

ANALYTIC 



. iv. 



parabola; if <<!, an ellipse; and if <>!. an hyperbola. 



Each of these cases will now l>e separately consul. -ml. 



0) The parabola, <-l. If = 1, then FP : LP = I, 

 i.e., FP = LP for every position of 

 the tracing point,* hence the curve 

 passes through A, the point mid- 

 way between and F, but does not 

 again cross the principal axis (cf. 

 also equations (2), above). 



Moreover, when e = 1, equation (1) 

 becomes 



Fto.34. 



i.e.. 



(3) 



which is the equation of the parabola, the coordinate axes 

 U ing the principal axis of the curve and the directrix. 

 Equation (3) shows that there is no point of this ]>;ir;il><>I;i 



JL 



for which #<-, and also that y changes from to 00 

 2t 



k 



when x increases from - to oo ; hence the parabola recedes 



2 



indefinitely from both axes in the first and fourth quadrants. 

 Its form is given in Fig. 34. 



(2) The ellipse, e<\. Equation (1) may be written in 

 the form 



This property enables one to construct any number of points lyiritf on the 

 parabola, thus: with F as center, and any radius not leas than \ OF, describe 

 a circle, then draw a line parallel to O Y and at a distance from it equal to 

 the chosen radius ; the points in which this line cuts the circle are points on 

 the parabola. Other points can be located in the same way. See also Note 

 II. ApiM-ndix. 



' .'liiation (4) enables one to construct any number of points on the 



