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The line joining the points $(2,1,3)$ and $(4,-2,5)$ cuts the plane $2 x+y-z=3 .$ What is the ratio in which the plane divideds the line?
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Let the ratio is $\mathrm{k}: 1$
Then, $0=\frac{4 k+2}{k+1}$
$\Rightarrow 4 k+2=0 \Rightarrow k=-\frac{1}{2}$
and $4=\frac{-2 \mathrm{k}+1}{\mathrm{k}+1} \Rightarrow 4 \mathrm{k}+4=-2 \mathrm{k}+1 \Rightarrow \mathrm{k}=-\frac{1}{2}$
Ilcnce, plane divides the line in ratio 1 : 2 cxternally.
Then, $0=\frac{4 k+2}{k+1}$
$\Rightarrow 4 k+2=0 \Rightarrow k=-\frac{1}{2}$
and $4=\frac{-2 \mathrm{k}+1}{\mathrm{k}+1} \Rightarrow 4 \mathrm{k}+4=-2 \mathrm{k}+1 \Rightarrow \mathrm{k}=-\frac{1}{2}$
Ilcnce, plane divides the line in ratio 1 : 2 cxternally.
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