If the square of the shortest distance between the lines (x-2)/(1)=(y-1)/(2)=(z+3)/(-3) an…

Question

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 $\frac{m}{n}$, where $m$, $n$ are coprime numbers, then $m+n$ is equal to :

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}\end{aligned}$$\right| \
\& =2 \hat{\mathrm{i}}-\hat{\mathrm{j}} \
\& \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
\]

Solutions · 1