Question
The square of the distance of the point of intersection of the line $\overrightarrow{r} = (\widehat{i} + \widehat{j} - \widehat{k}) + \lambda(\widehat{a} - \widehat{j})$, $a \neq 0$ and $\overrightarrow{r} = (4\widehat{i} - \widehat{k}) + \mu(2\widehat{i} + a\widehat{k})$ from the origin is :
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Solution
Sol. [Key Concept: Equate the two lines and match components to find $a$ and the point of intersection, then take its squared distance from the origin.]
Equating both the equations
$${(\widehat{i} + \widehat{j} - \widehat{k}) + \lambda(a\widehat{i} - \widehat{j}) = (4\widehat{i} - \widehat{k}) + \mu(2\widehat{i} + a\widehat{k}) }{1 + a\lambda = 4 + 2\mu }{1 - \lambda = 0 \Rightarrow \lambda = 1 }{- 1 = - 1 + a\mu }{a\mu = 0 \Rightarrow \mu = 0 }{a = 3 }$$point of intersection $= \widehat{i} + \widehat{j} - \widehat{k} + 3\widehat{i} - \widehat{j} = 4\widehat{i} - \widehat{k}$
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