Circle And System Of Circles Question 138
Question: A circle touches x-axis and cuts off a chord of length 2l from y-axis. The locus of the centre of the circle is
Options:
A) A straight line
B) A circle
C) An ellipse
D) A hyperbola
Show Answer
Answer:
Correct Answer: D
Solution:
If the circle $ x^{2}+y^{2}+2gx+2fy+c=0 $ touches the x-axis, then $ -f=\sqrt{g^{2}+f^{2}-c}\Rightarrow g^{2}=c $ …..(i) and cuts a chord of length 2l from y-axis
$ \Rightarrow 2\sqrt{f^{2}-c}=2l\Rightarrow f^{2}-c=l^{2} $ ………….(ii) Subtracting (i) from (ii), we get $ f^{2}-g^{2}=l^{2} $ .
Hence the locus is $ y^{2}-x^{2}=l^{2} $ , which is obviously a hyperbola.
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