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JEE Main 2019 · 10 Jan · Shift I

JEE Main 2019 Mathematics question on Straight Lines and Pair of Lines

A point PP moves on the line 2x−3y+4=02x - 3y + 4 = 0. If Q(1,4)Q(1, 4) and R(3,−2)R(3, -2) are fixed points, then the locus of the centroid of △PQR\triangle PQR is a line
  1. A.
    with slope 23\dfrac{2}{3}
  2. B.
    with slope 32\dfrac{3}{2}
  3. C.
    parallel to YY-axis
  4. D.
    parallel to XX-axis
Show answer and solution

Correct option: A

Solution

Let P = (x₀, y₀) with 2x₀ - 3y₀ + 4 = 0. The centroid G = (h, k) satisfies:

h = (x₀ + 1 + 3)/3 → x₀ = 3h - 4
k = (y₀ + 4 - 2)/3 → y₀ = 3k - 2

Substituting into the constraint: 2(3h - 4) - 3(3k - 2) + 4 = 0 → 6h - 8 - 9k + 6 + 4 = 0 → 6h - 9k + 2 = 0.

This is a line with slope 6/9 = 2/3.

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