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Question

# Let A(2,−3) and B(−2,1) be two angular points of △ABC. If the centroid of the triangle moves on the line 2x+3y=1, then the locus of the angular point C is given by

A
2x+3y=9
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B
2x3y=9
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C
3x+2y=5
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D
3x2y=3
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Solution

## The correct option is A 2x+3y=9A(2,−3),B(−2,1) are two angular points of △ABCLet C(α,β) be the third angular point of △ABCLet G be the centroid of △ABCThen, G=(2−2+α3,−3+1+β3) =(α3,β−23) .... (i)Since, G moves on the line 2x+3y=1Points on this line are of the form (x,1−2x3)⟹G=(t,1−2t3)From (i)α3=t and β−23=1−2t3⟹α=3t and β=3−2t ∴ C≡(3t,3−2t) Locus of point C is given by (3t−2)2+(3−2t+3)3=(3t+2)2+(3−2t−1)2⟹9t2+4−12t+36+4t2−24t=9t2+4+12t+4+4t2−8t⟹40t=32⟹t=45∴C≡(125,75)The equation of locus of point C will be parallel to line 2x+3y=1, sice centroid G lie on this line.∴ Equation of locus of C is given by 2x+3y=d Substituting C in the given equation 2x+3y=d we get d=9Thus, the locus of C is 2x+3y=9.

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