Question

If the shortest distance between the lines ${\displaystyle {\overrightarrow{r}}_1=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k})}$, λ∈R, α > 0 ${\displaystyle {\overrightarrow{r}}_2=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k})}$, μ∈R is 9, then α is equal to ______.

Options

• 3

• 4

• 5

• 6

Answer 1

If the shortest distance between the lines ${\displaystyle {\overrightarrow{r}}_1=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k})}$, λ∈R, α > 0 ${\displaystyle {\overrightarrow{r}}_2=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k})}$, μ∈R is 9, then α is equal to 6.

Explanation:

Given the equation of lines

${\displaystyle {\overrightarrow{r}}_1=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k})}$

${\displaystyle {\overrightarrow{r}}_2=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k})}$

Shortest distance = ${\displaystyle \left|\frac{({\overrightarrow{a}}_1\times {\overrightarrow{a}}_2).({\overrightarrow{b}}_1\times {\overrightarrow{b}}_2)}{|{\overrightarrow{b}}_1\times {\overrightarrow{b}}_2|}\right|}$

∴ 9 = ${\displaystyle \left|\frac{((\alpha +4)\hat{i}+2\hat{j}+3\hat{k}).(8\hat{i}+8\hat{j}+4\hat{k})}{\sqrt{64+64+16}}\right|}$

⇒ ${\displaystyle \left|\frac{8(\alpha +4)+16+12}{12}\right|}$ = 9

∴ α = 6, as α > 0