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Question

Prove that the locus of the centroid of the triangle whose vertices are (acost,asint,)(bsint,bcos),and (1,0),where t is a parameter, is circle.

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Solution

Let locus of centroid be at (h,k)
then
h=1+acost+bsint3
k=asintbcost3
3h1=acost+bsint ______ (1)
3k=asintbcost __________ (2)
square both equations and add
(3h1)2+(3k)2=a2(cos2t+sin2t)+b2(sin2t+cos2t)
(3h1)2+(3k)2=a2+b2
put h=x,k=y
(3x1)2+9y2=a2+b2
it is a circle
with center (13,0) and radius a2+b29=a2+b23.

1072965_1159555_ans_9f2dd7cc349b43eeb944b3764c2595db.png

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