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Prove that the locus of the centroid of ...

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

Text Solution

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Let (h,k) be the centroid. Then,
`h=(a cos t +b sin t +1)/(3)`
and `k=(a sin t -b cos t)/(3)`
or `3h-1=a cos t+b sin t ` (1)
and `3k=a sin t -b sin t ` (2)
Squaring and adding (1) and (2), we get
`(3h-1)^(2)+(3k)^(2)=a^(2)+b^(2)`
Hence, the locus of (h,k) is `(3x-1)^(2)+(3y)^(2)=a^(2)+b^(2)`
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