The lines (x - 2)/(2) = (y)/(- 2) = (z - 7)/(16) and (x + 3)/(4) =...

Given: $\dfrac{x - 2}{2} = \dfrac{y}{- 2} = \dfrac{z - 7}{16}$ and $\dfrac{x + 3}{4} = \dfrac{y + 2}{3} = \dfrac{z + 2}{1}$

$⇒ \dfrac{x - 2}{1} = \dfrac{y}{- 1} = \dfrac{z - 7}{8} = \lambda\left(\text{let}\right)$ and $\dfrac{x + 3}{4} = \dfrac{y + 2}{3} = \dfrac{z + 2}{1} = k\left(\text{let}\right)$

$⇒ x = \lambda + 2 = 4k - 3, y = - \lambda = 3k - 2$ and $z = 8\lambda + 7 = k - 2$

$⇒ \lambda = - 5 + 4k, \lambda = - 3k + 2, 8\lambda - k = - 9$

$⇒ - 5 + 4k = - 3k + 2$

$⇒ 7k = 7$

$⇒ k = 1, \lambda = - 3\left(1\right) + 2 = - 1$

$⇒ x = 1, y = 1, z = - 1$

$⇒ P ≡ (1,1,-1)$

Projection of PQ ≡ 2\overset{^}{i} - 2\overset{^}{k} on $QR$ line with direction ratio 2\overset{^}{i} + 3\overset{^}{j} + \overset{^}{k} is given by, 

$QR = \dfrac{4 - 2}{\sqrt{4 + 9 + 1}} = \dfrac{2}{\sqrt{14}}$

Now, in triangle $PQR$ we get,

$∴ {l}^{2} = 8 - \dfrac{4}{14} = \dfrac{108}{14}$

$⇒ 14{l}^{2} = 108$