The coplanar points $A, B, C, D$ are $(2 - x, 2, 2), (2, 2 - y, 2), (2, 2, 2 - z)$ and $(1, 1, 1)$ respectively, then

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1

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x + y + z = 1

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\frac{1}{1 - x} + \frac{1}{1 - y} + \frac{1}{1 - z} = 1

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None of these

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Solution

The correct option is A $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$
We have four coplanar points.
The three vectors connecting two of them at a time are thus coplanar.

$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} - x & y & 0 - x & 0 & z 1 - x & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow - x (- z) - y (- x - z + x z) + 0 = 0$

$\Rightarrow x z + x y + y z = x y z$

$\Rightarrow \frac{1}{y} + \frac{1}{z} + \frac{1}{x} = 1$

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The coplanar points $A, B, C, D$ are $(2 - x, 2, 2), (2, 2 - y, 2), (2, 2, 2 - z)$ and $(1, 1, 1)$ respectively, then