In the triangle with vertices at A (6, 3), B ( -6, 3), C (-6, -3) the median through A meets BC at P, the line AC meets the x-axis at Q, while R and S respectively denote the orthocenter and centroide of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is
$\begin{array}{l} L i s t - I L i s t - I I \\ i) P A) (0, 0) \\ i i) Q B) (6, 0) \\ i i i) R C) (- 2, 1) \\ i v) S D) (- 6, 0) \\ E) (6, - 3) \\ F) (- 6, 3) \end{array}$

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$D B E C$

$B A F C$

$D A F C$

$D A E C$

answer is C.

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Detailed Solution

Given A (6, 3) B (-6, 3) C (-6, -3)

Slope of AB = 0, Slope of BC $= \frac{- 3 - 3}{- 6 + 6} = \infty$

$\begin{array}{l} ∴ \bar{A B} ⊥ \bar{B C} \Rightarrow B = 90^{0} \\ ∴ R = o r t h o c e n t r e = B = (- 6, 3) \\ P = m i d p o int o f B C \\ = (\frac{- 6 - 6}{2}, \frac{3 - 3}{2}) = (- 6, 0) \\ E q u a t i o n o f \bar{A C} i s \\ y - 3 = \frac{- 3 - 3}{- 6 - 6} (x - 6) \\ \Rightarrow y - 3 = \frac{1}{2} (x - 6) \\ \Rightarrow 2 y - 6 = x - 6 \\ \Rightarrow x - 2 y = 0 \end{array}$

$\begin{array}{l} I t m e e t s X - a x i s \Rightarrow S u b y = 0 \Rightarrow x = 0 \\ ∴ Q = (0, 0) \\ S = (\frac{6 - 6 - 6}{3}, \frac{3 + 3 - 3}{3}) = (- 2, 1) \end{array}$