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Question

Consider the circle x2+y2=a2. Let A(a,0) and D be a given interior point of the circle. If BC be an arbitrary chord of the circle through point D, then the locus of the centroid of ΔABC is
  1. a circle whose radius is less than 2a3 units
  2. a circle whose radius is greater than 2a3  units
  3. a circle whose radius is equal to 2a3  units
  4. None of the above


Solution

The correct option is A a circle whose radius is less than 2a3 units
B,C lies on the circle 
parametric coordinate of 
B(acosα,asinα),C(acosβ,asinβ)
where α,β, (αβ) are parametric angles
and A(a,0)
Let the coordinates of centriod of ABC be G(h,k)
h=a+acosα+acosβ3,k=asinα+asinβ3
(ha3)2+k2=a29(1+1+2cos(αβ))
Hence locus will be 
(xa3)2+y2=a29(2+2cos(αβ))
Hence locus will be circle and radius will be 
<2a/3  (2+2cos(αβ)<4) 

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