Question

What is the condition for the circle $x^2+y^2+2gx+2fy+c=0$ to cut the x-axis at more than one point.

• A. c>g²
• B. c>f²
• C. f²>c
• D. g²>c

Answer 1

The correct option is D

g²>c

When the circle cuts the x-axis, the y-coordinate will be zero. If we substitute $y=0$ in the circles equation, we will get a quadratic in x. The roots of this equation are the points at which the circle cut the x-axis. So, to cut at more than one point, the discriminant of that quadratic should be greater than zero.
$x^2+y^2+2gx+2fy+c=0$
$y=0\Rightarrow x^2+2gx+c=0$
Discriminant $>0$

$\Rightarrow (2g{)}^2-4c>0$

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

$\Rightarrow g^2>c$