Let the image of the point ((5)/(3), (5)/(3), (8)/(3)) in the plane...

The image of the point $\left(\dfrac{5}{3}, \dfrac{5}{3}, \dfrac{8}{3}\right)$ w.r.t plane will be,

$\dfrac{x - \dfrac{5}{3}}{1} = \dfrac{y - \dfrac{5}{3}}{- 2} = \dfrac{2 - \dfrac{8}{3}}{1} = \dfrac{- 2\left(1 \times \dfrac{5}{3} - 2 \times \dfrac{5}{3} + \dfrac{8}{3} - 2\right)}{{1}^{2} + {2}^{2} + {1}^{2}}$

$⇒ \dfrac{x - \dfrac{5}{3}}{1} = \dfrac{y - \dfrac{5}{3}}{- 2} = \dfrac{2 - \dfrac{8}{3}}{1} = \dfrac{- 2 ( - 1 )}{6}$

$⇒ P ( x, y, z ) = ( 2, 1, 3 )$

But $PQ = 13$

${4}^{2} + {3}^{2} + ( \alpha - 3{)}^{2} = 169$

$| \alpha - 3 | = 12$

$\alpha = 15$ Since \left(\alpha > 0\right)

Hence this is the required answer.