Let A BC be a triangle formed by the lines 7 x − 6 y + 3 = 0 , x + 2 y − 31 = 0 and 9 x − 2 y − 19 = 0 . Let the point ( h , k ) be the image of the centroid of Δ A BC in the line 3 x + 6 y − 53 = 0 .

Question

Let A BC be a triangle formed by the lines 7 x − 6 y + 3 = 0 , x + 2 y − 31 = 0 and 9 x − 2 y − 19 = 0 . Let the point ( h , k ) be the image of the centroid of Δ A BC in the line 3 x + 6 y − 53 = 0 . Then h 2 + k 2 + hk is equal to:

TestHub · Accepted Answer

∴ centroid of △ ABC ​ = ( 3 9 + 3 + 5 ​ , 3 11 + 4 + 13 ​ ) = ( 3 17 ​ , 3 28 ​ ) ​ Let image of centroid with respect to line mirror is ​ ( h , k ) ∴ ( h − 3 17 ​ k − 3 28 ​ ​ ) ( − 2 1 ​ ) = − 1 &3 ( 2 h + 3 17 ​ ​ ) + 6 ⋅ ( 2 3 k + 28 ​ ​ ) = 53 ​ Solving (1) \& (2) we get h = 3 , k = 4 ∴ h 2 + k 2 + hk = 37

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