If the shortest distance between the lines x + 6 2 = y - 6 3 = z -...

Given equations are

$\dfrac{x - \lambda}{3} = \dfrac{y - 2\sqrt{6}}{4} = \dfrac{z + 2\sqrt{6}}{5}$ is passing through a point \left(\lambda, 2\sqrt{6}, 3\sqrt{6}\right)$and it's direction ratios are$3, 4, 5$.</p><p>So, </p><p>$\overset{\to}{a}1 = \lambda\overset{^}{i} + 2\sqrt{6}\overset{^}{j} - 2\sqrt{6}\overset{^}{k}$</p><p>$\overset{\to}{b}1 = 3\overset{^}{i} + 4\overset{^}{j} + 5\overset{^}{k}$</p><p>And,</p><p>$\dfrac{x + \sqrt{6}}{2} = \dfrac{y - \sqrt{6}}{3} = \dfrac{z - \sqrt{6}}{4}$is passing through a point \left(- \sqrt{6}, \sqrt{6}, \sqrt{6}\right)$ and it's direction ratios are $1, 2, 3$.

\overset{\to}{a}2 = - \sqrt{6}\overset{^}{i} + \sqrt{6}\overset{^}{j} + \sqrt{6}\overset{^}{k}

\overset{\to}{b}2 = 2\overset{^}{i} + 3\overset{^}{j} + 4\overset{^}{k}

Now,

\overset{\to}{a}2 - \overset{\to}{a}1 = \left(- \sqrt{6}\overset{^}{i} + \sqrt{6}\overset{^}{j} + \sqrt{6}\overset{^}{k}\right) - \left(\lambda\overset{^}{i} + 2\sqrt{6}\overset{^}{j} - 2\sqrt{6}\overset{^}{k}\right) = - \left(\sqrt{6} + \lambda\right)\overset{^}{i} - \sqrt{6}\overset{^}{j} + 3\sqrt{6}\overset{^}{k}

\overset{\to}{b}1 \times \overset{\to}{b}2 = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 3 & 4 & 5 \\ 2 & 3 & 4\end{cases}\right|

⇒ \overset{\to}{b}1 \times \overset{\to}{b}2 = \left(\overset{^}{i} - 2\overset{^}{j} + \overset{^}{k}\right)

So,

$⇒ \left|\overset{\to}{b}1 \times \overset{\to}{b}2\right| = \sqrt{6}$

We know that the shortest distance between two skew lines is $\dfrac{\left|\left(\overset{\to}{a}2 - \overset{\to}{a}1\right) \cdot \left(\overset{\to}{b}1 \times \overset{\to}{b}2\right)\right|}{\left|\overset{\to}{b}1 \times \overset{\to}{b}2\right|} = 6$

⇒ \left|\dfrac{\left(\left(- \sqrt{6} - \lambda\right)\overset{^}{i} - \sqrt{6}\overset{^}{j} + 3\sqrt{6}\overset{^}{k}\right) \cdot \left(\overset{^}{i} - 2\overset{^}{j} + \overset{^}{k}\right)}{\sqrt{6}}\right| = 6

$⇒ \left|\dfrac{4\sqrt{6} - \lambda}{\sqrt{6}}\right| = 6$

$⇒ \lambda = 10\sqrt{6}, - 2\sqrt{6}$

So, required sum is

$= {\left(10\sqrt{6} - 2\sqrt{6}\right)}^{2} = 384$