If $A (5, - 2), B (2, 4)$ and point C lies on the locus $y = 1 + x + x^{2}$ , find the equation of the locus of centroid of $△ A B C$ .

9 x^{2} - 39 x - 3 y + 41 = 0

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9 x^{2} - 39 x + 3 y + 40 = 0

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9 x^{2} - 39 x + 5 y + 41 = 0

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9 x^{2} - 39 x + 8 y + 41 = 0

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Solution

The correct option is B $9 x^{2} - 39 x - 3 y + 41 = 0$
A(5,-2)
B(2,4)
Let coordinates of C be $(α, 1 + α + α^{2})$
Centrioid(h,k)
$(α, 1 + α + α^{2}) h = \frac{5 + 2 + α}{3} α = 3 h - 7 α = \frac{- 1 + 4 + 1 + α + α^{2}}{3} 3 k = 3 + (3 h - 7) + {(3 h - 7)}^{2} 3 k = 3 + 3 h - 7 + 9 h^{2} - 42 h + 49 9 h^{2} - 39 h - 3 k + 45 = 0 9 x^{2} - 39 x - 3 y + 45 = 0$

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