question_answer

A variable circle having fixed radius 'a', passes through origin and meets the co-ordinate axes in points A and B. Locus of centroid of triangle OAB, where 'O' being the origin, is -

A) \[9\left( {{x}^{2}}+{{y}^{2}} \right)=4{{a}^{2}}\]

B)        \[9\left( {{x}^{2}}+{{y}^{2}} \right)={{a}^{2}}\]

C) \[9\left( {{x}^{2}}+{{y}^{2}} \right)=2{{a}^{2}}\]

D)        \[9\left( {{x}^{2}}+{{y}^{2}} \right)=8{{a}^{2}}\]

Correct Answer: A

Solution :

[a] Equation of circle \[\left( x-\alpha  \right)\left( x-0 \right)+\left( y-0 \right)\left( y-\beta  \right)=0\] \[{{x}^{2}}+{{y}^{2}}-\alpha x-\beta y=0\] \[a=\sqrt{\frac{{{\alpha }^{2}}}{4}+\frac{{{\beta }^{2}}}{4}}\] \[{{\alpha }^{2}}+{{\beta }^{2}}=4{{a}^{2}}\] Let centroid be (x, y) \[x=\alpha /3,\text{ }y=\beta /3\] \[\alpha =3x,\text{ }\beta =3y\] \[\therefore \text{ }locus\text{ }9\left( {{x}^{2}}+{{y}^{2}} \right)=4{{a}^{2}}\]