JEE MAIN - Mathematics (2023 - 25th January Evening Shift - No. 7)
The shortest distance between the lines $$x+1=2y=-12z$$ and $$x=y+2=6z-6$$ is :
3
$$\frac{5}{2}$$
$$\frac{3}{2}$$
2
Explanation
$L_{1}: \frac{x+1}{1}=\frac{y}{\frac{1}{2}}=\frac{z}{\frac{-1}{12}}$
$$ \begin{aligned} & L_{2}: \frac{x}{1}=\frac{y+2}{1}=\frac{z-1}{\frac{1}{6}} \\\\ & \text { S.D }=\left|\frac{(-\hat{i}+2 \hat{j}-\hat{k}) \cdot(2 \hat{i}-3 \hat{j}+6 \hat{k})}{7}\right| \\\\ & =\left|\frac{-2-6-6}{7}\right|=2 \text { units } \end{aligned} $$
$$ \begin{aligned} & L_{2}: \frac{x}{1}=\frac{y+2}{1}=\frac{z-1}{\frac{1}{6}} \\\\ & \text { S.D }=\left|\frac{(-\hat{i}+2 \hat{j}-\hat{k}) \cdot(2 \hat{i}-3 \hat{j}+6 \hat{k})}{7}\right| \\\\ & =\left|\frac{-2-6-6}{7}\right|=2 \text { units } \end{aligned} $$
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