If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\mathrm{m}, \mathrm{n}$ are coprime numbers, then $\mathrm{m}+\mathrm{n}$ is equal to :

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Aicharya · Accepted Answer

$\begin{aligned} & \overrightarrow{\mathrm{a}}=(2,1,-3) \\ & \overrightarrow{\mathrm{b}}=(-1,-3,-5) \\ & \overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 1 & 2 & -3 \\ 2 & 4 & -5\end{array}\right| \\ & =2 \hat{\mathrm{i}}-\hat{\mathrm{j}}\end{aligned}$
$\begin{aligned} & \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}=-3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\ & \mathrm{S}_{\mathrm{d}}=\frac{|(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}) \cdot(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}})|}{|\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}|}- \\ & =\frac{2}{\sqrt{5}} \\ & \left(\mathrm{~S}_{\mathrm{d}}\right)^2=\frac{4}{5} \\ & \mathrm{~m}=4, \mathrm{n}=5 \Rightarrow \mathrm{~m}+\mathrm{n}=9\end{aligned}$ ^

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and…

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