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 xcos⁡θ=ysin⁡θ=r, where θ=π4For point p we have r=62.Therefore, coordinates of Pare given by xcos⁡π4=−ysin⁡π4=62⇒x=6,y=6Since P(6, 6) lies on x2+y2+2gx+2fy+c=0∴ 72+12(g+f)+c=0Since y = x touches the circle. Therefore, the equation2x2+2x(g+f)+c=0⇒ 4(g+f)2=8c⇒(g+f)2=2c[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