question_answer

If a circle of constant radius 3k passes through the origin O and meets the coordinate axes at A and B, then the locus of the centroid of triangle OAB is

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

B) \[{{x}^{2}}+{{y}^{2}}={{(3k)}^{2}}\]

C) \[{{x}^{2}}+{{y}^{2}}={{(4k)}^{2}}\]

D) \[{{x}^{2}}+{{y}^{2}}={{(6k)}^{2}}\]

Correct Answer: A

Solution :

[a] Let the centroid of triangle OAB be (p, q). Hence, points A and B are (3p, 0) and (0, 3q), respectively. But diameter of circle, AB=6k Hence, \[\sqrt{9{{p}^{2}}+9{{q}^{2}}}=6k\] Therefore, the locus of (p, q) is \[{{x}^{2}}+{{y}^{2}}=4{{k}^{2}}\].