Let $l_{1}$ and $l_{2}$ be the lines $\bar{r_{1}} = λ (\hat{i} + \hat{j} + \hat{k})$ and $\bar{r_{2}} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})$ respectively. Let X be the set of all the planes H that contain the line $l_{1}$ . For a plane H, let d(H) denote the smallest possible distance between the points of $l_{2}$ and H. Let $H_{0}$ be a plane in X for which $d (H_{0})$ is the maximum value of d(H) as H varies over all planes in X. The value of $d (H_{0})$ is ______

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$\frac{1}{\sqrt{2}}$

$\frac{1}{\sqrt{3}}$

$\sqrt{3}$

$\sqrt{2}$

answer is C.

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Detailed Solution

$l_{1} : \bar{r_{1}} = λ (\hat{i} + \hat{j} + \hat{k}) l_{2} : \bar{r_{2}} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})$

Let the system of planes are ax + by + cz = 0 ______ (1)
It contains $l_{1} \Rightarrow a + b + c = 0$ ______ (2)
For largest possible distance between plane (1) and $l_{2}$
$\Rightarrow$ The line $l_{2}$ must be parallel to the palne
$\Rightarrow$ a + c = 0 ______ (3)
(2), (3) $\Rightarrow$ b = 0, a = – c
Equation of plane $H_{0}$ is x – z = 0
$d (H_{0}) = | \frac{0 + 1}{\sqrt{2}} | = \frac{1}{\sqrt{2}}$