The point (x1,y1) lies inside the circle x2+y2+2gx+2fy+c=0, if x21+y21+2gx1+2fg1+c is less than zero
True
From the figure we can see that, PC<r. this is the condition for P to lie inside the circle.
The coordinates of P are (x1,y1) and
C≡(−g,−f)
r=√g2+f2−c
PC<r
⇒√(x1+g)2+(y1+f)2<√g2+f2−c
⇒x21+y21+2gx1+2fy1+c<0
⇒ The given statement is correct