The circle x2 + y2 + 2gx + 2 fy + c = 0 does not intersect x - axis, if
Given that the circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x - axis. We need to find the relation between g and c.
We know that the value of y on the x - axis is 0.
⇒ x2 + (0)2 + 2gx + 2f(0) + c = 0
⇒ x2 + 2gx + c = 0
The quadratic equation will have no roots if the circle does not intersect the x - axis.
We know that for the quadratic equation ax2 + bx + c = 0 the condition to be satisfied for the equal roots is b2 - 4ac<0
⇒ (2g)2 - 4(1)(c)<0
⇒ 4(g2 - c)<0
⇒ g2 - c<0
⇒ g2<c
∴The correct option is (a).