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Centroid

A1, 4 and B...

Question

$A (1, 4) a n d B (6, - 3)$ are two vertices of ABC and C is a point on the locus $5 x + 4 y = 9$ . Find the equation of the locus of the centroid of $Δ A B C$ .

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Solution

Vertices of $△ A B C$ => $A (1, 4), B (6, - 3)$ and C is point on $5 x + 4 y = 9$
let centroid be $(h, k)$ = $(\frac{1 + 6 + x}{3}, \frac{4 - 3 + y}{3})$
$(x, y) = (3 h - 7, 3 k - 1)$ lies on $5 x + 4 y = 9$
$=> 5 (3 h - 7) + 4 (3 k - 1) = 9 => 15 h + 12 k - 30 = 0 => h + \frac{4}{5} k = 2$
locus of centroid => $x + \frac{4}{5} y = 2$

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