question_answer

A circle is drawn to cut a chord of length 2a unit along x-axis and to touch they-axis. The locus of the centre of the circle is:

A)  \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]                             

B)  \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]

C)  \[x+y={{a}^{2}}\]           

D)         \[{{x}^{2}}-{{y}^{2}}=4{{a}^{2}}\]

E)  \[{{x}^{2}}+{{y}^{2}}=4{{a}^{2}}\]

Correct Answer: B

Solution :

\[\because \]\[2\sqrt{{{g}^{2}}-c}=2a\] \[\Rightarrow \]               \[{{g}^{2}}-x={{a}^{2}}\]                               ?. (i) and            \[{{f}^{2}}=c\]                                 ...(ii) On solving Eqs. (i) and (ii), we get                 \[{{g}^{2}}-{{f}^{2}}={{a}^{2}}\] \[\therefore \]Locus of centre of the circle is \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]