1;<> XJV.i /.>'//' OJCOMBTBJ [('... 



on the radius of the circle, ami through l\ draw a line 

 perpendicular to OJ\ ; this line is tin- required polar. 



- milarly tin* pole may be constructed, if tin- polar and 

 the circle are givm. 



95. Circles through the intersections of two given circles. 



(iiven two circles whose equations are 



* + y> + :> <V + 2^,y + C, = 0, . . . (1) 

 and 2^ + ^+2 0,r + 2JF' s y + (7 2 = 0. . . . (2) 



These circles intersect, in general, in two finite points 

 />,=(./,, )/ l ) and PI=(XH y 2 )i an( l (Art. 41) the equation 



(7, 

 ->2<V + 2^+a) = 0, . . . ( 



where A: is any constant, represents a curve which passes 

 through these same points P l and P* 



The locus of equation (3) is, moreover, a circle ( Art. 7 

 hence, a series of different values being assigned to the param- 

 eter A:, equation (3) represents what is called a -family" 

 of circles; each one of these circles passing through the t\\<> 

 points l\ and P 2 in which the given circles (1) and 

 intersect each other. 



96. Common chord of two circles. If in equation < 

 Art. ( ."). the parameter k be given the particular vain-- 



- 1, the equation reduces to 



2(^,-^ 1 ^ + 2(JP l -l' 2 )y+(7 1 -Ci=0, ... (1) 



which is of the first degree, and therefore represent 

 straight line; but this locus belongs to the family repre- 

 sented by equation (3) of Art. 95, hence it passes through tin- 

 two points PI and P 2 in which the circle^ ( 1 > and ( J > inter- 

 sect. This line (4) is, therefore, the common chord* of 

 these circles. 



