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A plane meets the co-ordinate axes at the points $A, B, C$ respectively in such a way that the centroid of $\triangle A B C$ is $\left(1, r, r^{2}\right)$ for some real $r$. If the plane passes through the point $(5,5,-12)$ then $r=$
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Verified Answer
The correct answers are:
$\frac{3}{2}$, $-4$
Centroid is $\mathrm{G}(\mathrm{a} / 3, \mathrm{~b} / 3, \mathrm{c} / 3) \equiv\left(1, \mathrm{r}, \mathrm{r}^{2}\right)$
$a / 3=1 \quad \Rightarrow a=3, b=3 r, c=3 r^{2}$
Now,
$\begin{array}{l}
\frac{5}{a}+\frac{5}{b}-\frac{12}{c}=1 \\
\Rightarrow \frac{5}{3}+\frac{5}{3 r}-\frac{12}{3 r^{2}}=1 \\
\Rightarrow \frac{5 r^{2}+5 r-12}{3 r^{2}}=1 \\
\Rightarrow 5 r^{2}+5 r-12=3 r^{2} \\
\Rightarrow 2 r^{2}+5 r-12=0 \\
\Rightarrow r=\frac{-5 \pm \sqrt{25+96}}{4} \\
\Rightarrow r=\frac{-5 \pm 11}{4}=-4,3 / 2
\end{array}$
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