n<- 01 <M// //;> vn. 



NOTE 1. This circle is real only if G* + F* - C> ; for, if 

 <?i+F* -C<0, 



its square root is imaginary, and no real values of x and y can then sat MY 

 equation (J) ; while if<7 + F*-C = 0, then equation ( _' ) m luces to 



(*+<?)' + (y + F) = 



which may be called the equation of a "point circle/' since the coordi- 

 nates of but one real point, vi/. (-G, -F), will satisfy equation (:$). 

 If, however, G* + F* -C>0, then equation (1) represents a real circle 

 for all values of G, F, and C, subject to this single limitation. 



NOTE & I :\ery equation of the form Ax* +Ay* + 2Gx + 2 Fy + C = 

 represents a circle, for, by Art. 08, this equation has the same locus as 



C 1 F 1 ( J 



has x 2 + y* + 2 z + 2 y + =0, and this last equation is of the 



A A A 



form of equation (1). 



Hence^ interpreted in rectangular coordinates^ every equation 

 of the second degree from which the term in xy is absent^ and 

 in which the coefficient of x* equals that of y 2 , represents a 

 circle. 



80. Equation of a circle through three given points, 

 means of equation [31], or of the equation 



** + / + 20a? + 2-Fy+ (7=0, . . . (1) 



which has been shown in Art. 79 to be its equivalent, 

 the problem of finding the equation of a circle which shall 

 pass through any three given points not lying on a straight 

 line can be solved ; i.e., the constants A, A:, and r, or 6?, F, 

 ami C, may be so determined that the circle shall j.ass 

 through the three given points. 



To illustrate : let the given points be (1, 1), (2,~1), and 

 2), and let & + y 2 4- 2Gx + 2 Fy + C = be the equa- 

 tion of the circle that passes through these points; to iiixl 

 the values of the constants #, F, and 0. Since the point 

 (1, 1) is on this circle, therefore (cf. Art. 35), 



1+1 + 20 f-2J T +C r =0; 



