Conic Sections Question 309
Question: 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-
Options:
A) $ c>0 $
B) $ c<0 $
C) $ c=0 $
D) $ c\le 0 $
Show Answer
Answer:
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 $
BETA
