Let the equation ax ^2+2 hxy + by ^2+2 gx +2 fy + c =0 represent a point circle other than the origin. Then which one of the following conditions must hold?

Step-by-step solution

Given: a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 (i) If equation (i) represents a circle, then a=b & h=0 Then eq ^ n (i) reduces to: b x^2+b y^2+2 g x+2 f y+c=0 x^2+y^2+2 g b x+2 ( f b ) y+ c b =0 ...(ii) Equation (ii), represents point circle. Hence radius r=0 aligned & ( g b )^2+ ( f b )^2- c b =0 & c b = g^2 b^2 + f^2 b^2 & cb = g ^2+ f ^2 aligned cb >0 g^2>0 & f^2>0