Let the triangle PQR be the image of the triangle with vertices ( 1 , 3 ) , ( 3 , 1 ) and ( 2 , 4 ) in the line x + 2 y = 2 . If the centroid of △ PQR is the point ( α , β ) , then 15 ( α − β ) is equ

Question

Let the triangle PQR be the image of the triangle with vertices ( 1 , 3 ) , ( 3 , 1 ) and ( 2 , 4 ) in the line x + 2 y = 2 . If the centroid of △ PQR is the point ( α , β ) , then 15 ( α − β ) is equal to :

TestHub · Accepted Answer

The centroid G ′′ ( α , β ) of △ PQR be image of centroid of given triangle P ′ Q ′ R ′ . Centroid of Δ P ′ Q ′ R ′ = ( 3 1 + 3 + 2 ​ , 3 3 + 1 + 4 ​ ) = G ′ ( 2 , 3 8 ​ ) . Image of G ( 2 , 3 8 ​ ) , w.r.t. line x + 2 y = 2 is ( α , β ) Then 1 α − 2 ​ = 2 β − 3 8 ​ ​ = 1 + 4 − 2 ( 2 + 3 16 ​ − 2 ) ​ ∴ 1 α − 2 ​ = 2 β − 3 8 ​ ​ = − 15 32 ​ ∴ α = − 15 2 ​ and β = − 5 8 ​ Then 15 ( α − β ) = 15 ( − 15 2 ​ + 15 24 ​ ) = 22

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