Three mutually perpendicular· lines are drawn from the point $(1, 2, - 1) .$ If one of the lines is perpendicular to the $x$ -axis and the direction ratios of the second line are $(1, 2, - 1),$ then which of the following is/ are possibly the equation(s) of the third line?

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$\frac{x - 1}{5} = \frac{y - 3}{- 2} = \frac{z + 1}{1}$

none of these

$\frac{x + 4}{5} = \frac{y - 4}{- 2} = \frac{z + 2}{1}$

$\vec{r} = 6 \hat{i} + λ (5 \hat{i} - 2 \hat{j} + \hat{k})$

answer is A, C.

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Detailed Solution

Let the direction ratios of the first and third

lines be $(0, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2})$

Direction ratios of second line are 1, 2 -1.

Since lines are perpendicular,

$2 m_{1} - n_{1} = 0, l_{2} + 2 m_{2} - n_{2} = 0$

On solving, we get

$\Rightarrow \frac{l_{1}}{0} = \frac{m_{1}}{1} = \frac{n_{1}}{2} and \frac{l_{2}}{5} = \frac{m_{2}}{- 2} = \frac{n_{2}}{1}$

Hence, equation of the required line are given by

$\frac{x - 1}{5} = \frac{y - 2}{- 2} = \frac{z + 1}{1} = λ$

For $λ = 1 and λ = - 1, we get (a) and (c)$

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