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JEE MainMathematicsToughStraight Lines

A point P moves on the line 2x - 3y +4 = 0. If Q(1, 4) and R(3,
-2) are fixed points, then the locus of the centroid of Δ\DeltaPQR is
a line

Show answer and solution
Let Point P is (α\alpha, β\beta) and centroid of Δ\DeltaPQR
is (h, k), then 3h = α\alpha + 1 + 3 and 3k = β\beta + 4 - 2
⇒α\Rightarrow \alpha = 3h - 4 and β\beta = 3k - 2
Because (α\alpha, β\beta) lies on 2x - 3y + 4 = 0
⇒\Rightarrow 2(3h - 4) - 3(3k - 2) + 4 = 0
⇒\Rightarrow locus is 6x - 9y + 2 = 0 whose slope 23\frac{2}{3}

Source: Grass-G / M-22 Mathematics Paper 19 Dec Eng.docx

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