I.*, ANALYTIC GEOMETRY [Cm, IV. 



U known merely as the path traced l.\ a jM.int \vliicli in 

 nndei- k'iven ruiiilitioiis or laws. Such acur\ . for instance, 

 .< path of a cannon kill, or other projectile, moving 

 nnder tin- inlinence of a known initial force and tin- force of 

 gravity. Another such <-ur\r is that in which iron filings 

 arrange themselves when acted upon hy kn<>\\n magnetic 

 forces. The orbits of the planets and other astronomical 



168, acting under the iiifineiice of certain centers of f 

 are important examples of this class of " given lot i." 



In such problems as these, the method used in Arts. 4o to 45, 

 cannot, in griicral, he applied. A method that can often he 

 ei, ployed, after the construction of an appropriate figure, is: 



(1 ) From the figure, express the known law, under which 

 the point moves, by means of an equation involving g 

 metric magnitudes; this equation may be called tin 

 metric equation." 



Ueplace each geometric magnitude by its equivalent 



algebraic value, expressed in terms of the coordinates of 



moving point and given constants; then simplify this 



algebraic equation, and the result is the desired equation of 



the locus. 



47. Equation of a circle : second method. To illustrate 

 this second method of finding the equation of a locus, con- 

 sider the circle as the path traced lv a 

 point which moves so that it is always 

 at a given constant distance from a : 

 point. From this definition, find its 

 -. piation. 



^ Let C = (3, 2) be the given fixed 



point, and let P =(x< y) be a point that 

 moves so as to be always at the distance 2| from C. Then 

 CP = , . . . [geometric equation] 



u 



s, 



