A line with direction ratio 2, 1, 2 meets the lines x = y + 2 = z...

Let $P\left(t, t - 2, t\right)$ and $Q\left(2s - 2, s, s\right)$

$DR ' s of PQ$ are \left(2, 1, 2\right)$</p><p>$⇒ \dfrac{2s - 2 - t}{2} = \dfrac{s - t + 2}{1} = \dfrac{s - t}{2}$</p><p>$⇒ \dfrac{2s - 2 - t}{2} = \dfrac{s - t + 2}{1}$</p><p>$⇒ 2s - 2 - t = 2s - 2t + 4$</p><p>$⇒ t = 6$</p><p>$⇒ \dfrac{s - t + 2}{1} = \dfrac{s - t}{2}$</p><p>$⇒ \dfrac{s - 6 + 2}{1} = \dfrac{s - 6}{2}$</p><p>$⇒ 2s - 8 = s - 6$</p><p>$⇒ s = 2$</p><p>$⇒ P \left(6, 4, 6\right)$and$Q\left(2, 2, 2\right)$</p><p>$PQ: \dfrac{x - 2}{2} = \dfrac{y - 2}{1} = \dfrac{z - 2}{2} = \lambda$</p><p>$⇒ x = 2\lambda + 2, y = \lambda + 2\text{and}z = 2\lambda + 2$</p><p>Let,$F ( 2\lambda + 2, \lambda + 2, 2\lambda + 2 )$</p><p><img alt="" src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/2acdd6e3-2ba4-4d2b-97d2-f229f5d6c906.png" style="width: 250px; height: 146px;"/></p><p>$A\left(1, 2, 12\right)$</p><p>$\overset{\to}{AF} \cdot \overset{\to}{PQ} = 0$</p><p>$⇒ 2\left(2\lambda + 1\right) + 1$\left(\lambda\right) + 2\left(2\lambda - 10\right) = 0$

$⇒ 4\lambda + 2 + \lambda + 4\lambda - 20 = 0$

$⇒ \lambda = 2$

So $F\left(6, 4, 6\right)$ and $AF = \sqrt{65}$

$⇒ {l}^{2} = 65$