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Question

If a variable circle having fixed radius a, passes through origin and meets the coordinates axes at point A and B respectively, then the locus of centroid of OAB, where O is the origin, is

A
9(x2+y2)=4a2
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B
9(x2+y2)=a2
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C
9(x2+y2)=2a2
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D
9(x2+y2)=8a2
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Solution

The correct option is A 9(x2+y2)=4a2
Let the variable circle be
x2+y2+2gx+2fy=0
Putting x=0
y=0,2fA=(0,2f)
Putting y=0
x=0,2gB=(2g,0)

Then radius of the circle is
r=g2+f2a2=g2+f2

Let P(h,k) be the centroid of OAB, we get
h=2g3,k=2f3g2+f2=a29h24+9k24=a2

Hence, the required locus is
9(x2+y2)=4a2

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