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Offline Centres

If three planes, $P_{1} : 2 x + y + z - 1 = 0,$ $P_{2} : x - y + z - 2 = 0,$ $P_{3} : α x - y + 3 z - 5 = 0$ intersects at $‘ P ’$ on $x o y$ plane, and at $Q$ on $y o z$ plane, $‘ O ’$ is origin. Then
LIST – I LIST – II
A) $α$ is i) 1/2
B) Drs of line perpendicular $P_{3}$ and passes through $P is a, b, c$ then $\frac{a + b + c}{a - c} =$ ii) 1
C) Length of projection of PQ on X-axis iii) 6
D) Centroid of $∆ O P Q$ is $(a, b, c) \to 3 a + 2 b + c$ iv) 4

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$A \to 1, B \to 3, C \to 4, d \to 2$

$A \to 4, B \to 2, C \to 3, d \to 1$

$A \to 4, B \to 3, C \to 2, d \to 1$

$A \to 3, B \to 4, C \to 1, d \to 2$

answer is C.

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Detailed Solution

Three planes meets at 2 points → they have ∞-solutions
$|\begin{array}{l} 2 1 1 \\ 1 - 1 1 \\ α - 1 3 \end{array}| = 0 \to α = 4$
$\begin{array}{l} P_{1} \to 2 x + y + z = 0 \\ P_{2} \to x - y + z = 2 \\ P_{3} \to 4 x - y + 3 z = 5 \end{array}}$ $\begin{array}{l} P = xoy plane = (1, - 1, 0) \\ Q = yoz plane = (0, \frac{- 1}{2}, \frac{3}{2}) \end{array}$
Line perpendicular to P3, passing ‘P’ $\to \frac{x - 1}{4}, \frac{y + 1}{- 1} = \frac{z}{3}$

$\frac{a + b + c}{a - c} =$
$P \bar{Q} = (1, \frac{- 1}{2}, \frac{- 3}{2})$ Projection of PQ on x-axis = 1
Centroid $ΔPQR = (\frac{1}{3}, \frac{- 1}{2}, \frac{1}{2})$

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