1 ''> ANALYTIC GEOMETRY [Cfc. VUL 



To derive the equation from this definition, let C=(h^ k) 

 be the center, r the radius, and P= (x, y) any point on i IK 



curve. Also draw the ordinal* s 

 Mfl and MP, and the line OR 

 parallel to the z-axis ; then 

 CP = r ; [geometric equation] 

 but (Ait. 26), 



hence V(* - A) 2 + (y - 





which is the equation of the circle whose radius is r, and 

 whose center has the coordinates h and /-. 



With given fixed axes, equation [31] may, by rightly 

 choosing A, A:, and r, represent any circle whatever; it is. 

 therefore, called the general equation of the circle. Of its 

 special forms one is, because of its very frequent applica-. 

 tion, particularly important ; this form is the one for wliii li 

 the center coincides with the origin : in that case h = k = 0, 

 and equation [31] becomes 



r*.t . . . [32] 



* Equation [31] may be written in the form 



the first member then becomes positive if the coordinates of any point n 

 of the circle are substituted for x and y, it becomes negative for a point inside 

 of the circle, and zero for a point on the circle. Hence the circle may be 

 regarded as the boundary which separates that part of the plane for which 

 the function (x A) 2 + (y - )* - r 2 is positive from the part for which this 

 function is negative. The inside of the circle may therefore be called its nega- 

 tive side, while the outside is its positive side (cf. foot-note, Art. 43). 



t If one is unrestricted in his choice of axes, then an equation of the form 

 of [32] may represent any circle whatever, the axes need merely be chosen 

 perpendicular to each other and through its center; equation [31] Is more 

 general in that, the rectangular axes being determined by other considera- 

 tions, it may still represent any circle whatever. 



