Question

The circle $x^2+y^2+2gx+2fy+c=0$ does not intersect x-axis, if

• A. $g^2<c$
  
• B. $g^2>c$
  
• C. $g^2>2c$
• D. none of these

Answer 1

The correct option is A

$g^2<c$

Given : $x^2+y^2+2gx+2fy+c=0\dots \left(1\right)$

The given circle intersects the x-axis

The equation of circle becomes

$x^2+2gx+c=0\dots \left(2\right)$

Solving equation (2) :

$\therefore$ Discriminant, $D=\sqrt{4g^2-4c\ge 0}$

$\Rightarrow 4g^2-4c\ge 0$

$\Rightarrow g^2-c\ge 0$

$\Rightarrow g^2\ge c$

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