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

Step-by-step solution

Let, coordinate of centroid is ( h ,   k ) and coordinate of point P is ( α ,   β ) We know that the centroid of a triangle with vertices x 1 ,   y 1 ,   x 2 ,   y 2 and x 3 ,   y 3 is x 1 + x 2 + x 3 3 ,   y 1 + y 2 + y 3 3 . Hence, in ∆ P Q R , we have α + 1 + 3 3 = h and β + 4 - 2 3 = k ⇒ α = 3 h - 4 , β = 3 k - 2 Since ( α ,   β ) lies on the line 2 x - 3 y + 4 = 0 Hence, 2 3 h - 4 - 3 3 k - 2 + 4 = 0 ⇒ 6 h - 9 k +