If the circle  x2+y2+2gx+2fy+c=0 is touched by y=x at P such that OP=62 , then the value  of c is

The equation of the line y=x in distance form is xcosθ=ysinθ=r , where θ=π4 For point P,r=62 , therefore coordinates of P are given by xcosπ4=ysinπ4=62 ⇒x=6,y=6 Since P(6,6) lies on x2+y2+2gx+2fy+c=0 72+12(g+f)+c=0 …….(1)Since y = x touches the circle , the equation 2x2+2x(g+f)+c=0 has equal rots⇒4(g+f)2=8c ⇒(g+f)2=2c ………….(2)From (1), we get [12(g+f)2]=[−(c+72)]2                         ⇒144(g+f)2=(c+72)2                        ⇒144(2c)=(c+72)2                       ⇒(c−72)2=0⇒c=72