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Equation of Tangent at a Point (x,y) in Terms of f'(x)

If A 5, -2 ...

Question

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 $Δ A B C$

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Solution

$A (5, - 2), B (2, 4)$ and C lies on $y - 1 = x + x^{2}$
let C be $(x, y)$
=>centroid of $△ A B C = (\frac{5 + 2 + x}{3}, \frac{- 2 + 4 + y}{3}) = (h, k) => (x, y) = (3 h - 7, 3 k - 2)$
$(x, y)$ lies on $y - 1 = x + x^{2} => 3 k - 2 - 1 = 3 h - 7 + (3 h - 7)^{2} => 3 k - 3 = 3 h - 7 + 9 h^{2} - 42 h + 49 => 9 h^{2} - 39 h - 3 k + 45 = 0$
locus => $x^{2} - 13 x - y + 15 = 0$

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