A) A straight line
B) A circle
C) An ellipse
D) A hyperbola
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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