The vertices A & B of a triangle ABC are (2, 5), (4, -11), C moves on the line L = 9x + 7y + 4 =0. Then the locus of the centroide of the triangle ABC is parallel to

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answer is D.

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Detailed Solution

Given A (2, 5) B ( 4, -11) and Let $C (α, β)$

P (x, y) be the centroide.

$\begin{array}{l} ∴ p (x, y) = (\frac{2 + 4 + α}{3}, \frac{5 - 11 + β}{3}) \\ (x, y) = (\frac{α + 6}{3}, \frac{β - 6}{3}) \\ \frac{α + 6}{3} = x, \frac{β - 6}{3} = y \\ α + 6 = 3 x, β - 6 = 3 y \\ α = 3 x - 6, β = 3 y + 6 \end{array}$

$\begin{array}{l} C (α, β) l i e s o n 9 x + 7 y + 4 = 0 \Rightarrow 9 α + 7 β + 4 = 0 \\ \Rightarrow 9 (3 x - 6) + 7 (3 y + 6) + 4 = 0 \\ \Rightarrow 27 x - 54 + 21 y + 42 + 4 = 0 \\ \Rightarrow 27 x + 21 y + 8 = 0 \\ S l o p e = - \frac{27}{21} \Rightarrow - \frac{9}{7} \\ a n d s l o p e o f 9 x + 7 y + 4 = 0 i s - \frac{9}{7} \\ ∴ T h e l o c u s i s p a r a l l e l t o L = 0 \end{array}$