If the origin and point $P (2, 3, 4), Q (1, 2, 3)$ and $R (x, y, z)$ are co-planar then

x - 2 y - z = 0

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x + 2 y + z = 0

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x - 2 y + z = 0

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2 x - 2 y + z = 0

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Solution

The correct option is C $x - 2 y + z = 0$
$O, P, Q, R$ are co-planar

$∴ [\to O R \to O P \to O Q] = 0$

$∣ ∣ ∣ ∣ \begin{matrix} x & y & z 2 & 3 & 4 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣$

Expanding the determinant, we get
$x (9 - 8) - y (6 - 4) + z (4 - 3) = 0$

$x - 2 y + z = 0$

Points $P, Q$ satisfy equations given in option C.

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