question_answer

Under which one of the following conditions does the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]meet the x-axis in two points on opposite sides of the origin?

A) \[c>0\]

B) \[c<0\]

C) \[c=0\] 

D) \[c\le 0\]

Correct Answer: B

Solution :

[b] For a circle to meet x-axis in two points on the opposite side of the origin its radius r, should be more the distance of its centre from the origin. Co-ordinate of centre of the circle \[{{x}^{2}}+{{y}^{2}}+2gx\] \[+2fy+c=0\] is \[(-g,-f):\] In the figure shown, \[OQ=OP=r,\] and distance of centre C, from origin, O is CO \[r>\sqrt{OC}\] i.e. \[r>\sqrt{{{(-g)}^{2}}+{{(-f)}^{2}}}\] or \[\sqrt{{{(-g)}^{2}}+{{(-f)}^{2}}-c}>\sqrt{{{(-g)}^{2}}+{{(-f)}^{2}}}\] or, \[{{g}^{2}}+{{f}^{2}}-c>{{g}^{2}}+{{f}^{2}}\] or, \[-c>0\] or, \[c<0\]