Question

The circle x² + y² + 2gx + 2fy + c = 0 does not intersect x-axis, if
(a) g² < c
(b) g² > c
(c) g² > 2c
(d) none of these

Answer 1

(a) $g^2<c$

Given:
x² + y² + 2gx + 2fy + c = 0 ...(1)

The given circle intersects the x-axis.
The equation of circle becomes x² + 2gx + c = 0. ...(2)

Solving equation (2):
∴ Discriminant, D = $\sqrt{4g^2-4c}\ge 0$
$$\begin{gathered} \Rightarrow 4g^2-4c\ge 0 \\ \Rightarrow g^2-c\ge 0 \\ \Rightarrow g^2\ge c \end{gathered}$$

Hence, if $g^2<c$, then the given circle will not intersect the x-axis.