60 



ANALYTIC GEOMETRY 



[Cm, in. 



hence the locus cuts each axis in one point onlv. .n.-i tli.it ]...int is the 

 origin. The equation may be written in tin* form y= V-l ./-. \\hi.h 

 shows that if x be negative y is imaginary ; hence there is no point of 

 this locus on the negative side of the y-axis. 



Again: for each positive value of x there are two real val 

 1 1 in ucrically equal, but opposite in sign; hence this locus passes through 

 the origin, lies wholly in the first and fourth quadrants, and is syrain* ti i- 

 cal with regard to the x-axis. 



The equation shows also that x may have any ]><iti\r value, however 

 great, and that y increases when r increases; these facts show th:r 

 locus recedes indefinitely from both axes, that it is an open curve of 

 one branch. It is called a parabola and has the form shown in 1 i-. 17. 

 DUcwuion of the equation x* + y 9 = a 1 . 



Intercepts : if x = 0, then y a, and 

 if y = 0, then x = a ; hence for each 

 axis there are two intercepts, each of length 

 a, and on opposite sides of the on 

 i.e., four positions of the tracing point are : 

 A=(a, 0), A' = (~a t 0), /* = (, a). 

 ' = (0, -vi). 



This equation may also be written 



y = Va - x*, 



which shows that every value of x gives 

 two corresponding values of y which are 

 numerically equal, but of opposite 

 the locus is, therefore, symmetrical with regard to the x-axis. It also 

 shows that, corresponding to any value of x numerically greater than a, 

 y is imaginary ; the tracing point, therefore, does not move further from 

 the y-axis than i a, i.e., further than the points A and A'. Moreover, 

 as x increases from to a, y remains real and changes gradually from 

 fa to 0, or from a to 0; i.e., the tracing point moves continuously 

 from B to A , or from Bf to A . 



Again, if x decreases from to - a, y remains real and changes con- 

 tinuously from + a to 0, or from a to 0; i.e., the tracing point moves 

 continuously from B to A' or from D' to A'. 



nilarly, the equation may be written x = Va* y a , which shows 

 that the curve is also symmetrical with regard to the y-axis. and that 

 til-- tracing point does not move farther than a from the x-axis. 



From these facts it follows that this locus is a closed curve of only one 

 branch. It is a circle of radius a, with its center at the origin ; this curve 

 will be studied in detail in Chap. VII. 



