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If the plane $7 x+11 y+13 z=3003$ meets the co-ordinate axes in $A, B, C$, then the centroid of the $\triangle A B C$ is
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Verified Answer
The correct answer is:
$(143,91,77)$
Given plane is $7 x+11 y+13 z=3003$
$$
\begin{array}{ll}
\Rightarrow & \frac{7 x}{3003}+\frac{11 y}{3003}+\frac{13 z}{3003}=1 \\
\Rightarrow & \frac{x}{429}+\frac{y}{273}+\frac{z}{231}=1
\end{array}
$$
This plane meets the coordinate axes.
The coordinates are
$$
\begin{aligned}
& A(429,0,0), B(0,273,0), C(0,0,231) \\
& \begin{aligned}
\text { The centroid of } \Delta A B C & =\left(\frac{429}{3}, \frac{273}{3}, \frac{231}{3}\right) \\
& =(43,91,77)
\end{aligned}
\end{aligned}
$$
$$
\begin{array}{ll}
\Rightarrow & \frac{7 x}{3003}+\frac{11 y}{3003}+\frac{13 z}{3003}=1 \\
\Rightarrow & \frac{x}{429}+\frac{y}{273}+\frac{z}{231}=1
\end{array}
$$
This plane meets the coordinate axes.
The coordinates are
$$
\begin{aligned}
& A(429,0,0), B(0,273,0), C(0,0,231) \\
& \begin{aligned}
\text { The centroid of } \Delta A B C & =\left(\frac{429}{3}, \frac{273}{3}, \frac{231}{3}\right) \\
& =(43,91,77)
\end{aligned}
\end{aligned}
$$
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