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If the origin and the points $(1,2,3),(2,3,4)$ and $(x, y, z)$ are coplanar, then
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Verified Answer
The correct answer is:
$x-2 y+z=0$
For give coplanar points, we write
$$
\begin{aligned}
&\left|\begin{array}{lll}
1 & 2 & 3 \\
2 & 3 & 4 \\
x & y & z
\end{array}\right|=0 \\
\therefore & x(8-9)-y(4-6)+z(3-4)=0 \\
\therefore &-x+2 y-z=0 \Rightarrow x-2 y+z=0
\end{aligned}
$$
$$
\begin{aligned}
&\left|\begin{array}{lll}
1 & 2 & 3 \\
2 & 3 & 4 \\
x & y & z
\end{array}\right|=0 \\
\therefore & x(8-9)-y(4-6)+z(3-4)=0 \\
\therefore &-x+2 y-z=0 \Rightarrow x-2 y+z=0
\end{aligned}
$$
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