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

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

$\vec{AB}= x\hat{i}- y \hat{j}, \vec{AC}= x\hat{i} - z\hat{k}, \vec{AD} = (x-1) \hat{i} - \hat{j} - \hat{k}$

Clearly, these  vectors are coplanar.

$∴  \begin{vmatrix}x & -y & 0\\x & 0 & -z\\x-1 & -1 & -1\end{vmatrix}=0⇒ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1$