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Centroid

Let α, β, γ b...

Question

Let $α, β, γ$ be the roots of the equation $x^{3} - 12 x^{2} + 44 x - 48 = 0.$ If the coordinates of the vertices of a triangle are $A (α, \frac{1}{α}),$ $B (β, \frac{1}{β})$ and $C (γ, \frac{1}{γ}),$ then the centroid of the $△ A B C$ is

(4, \frac{11}{36})

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(\frac{11}{36}, 4)

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(3, \frac{5}{11})

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(\frac{5}{11}, 3)

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Solution

The correct option is A $(4, \frac{11}{36})$
Given : $x^{3} - 12 x^{2} + 44 x - 48 = 0$

Centroid $= ⎛ ⎜ ⎜ ⎜ ⎝ \frac{α + β + γ}{3}, \frac{\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}}{3} ⎞ ⎟ ⎟ ⎟ ⎠$

Now, $\frac{α + β + γ}{3} = \frac{12}{3} = 4$

$\frac{1}{3} (\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}) = \frac{1}{3} (\frac{α β + β γ + γ α}{α β γ}) = \frac{1}{3} (\frac{44}{48}) = \frac{11}{36}$

Therefore, centroid $= (4, \frac{11}{36})$

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