The point P(a, b) undergoes the following three transformations...

Image of $$A\left(a, b\right)$ along $y = x$ is ${P}^{'}\left(b, a\right) .$

Translating it $2$ units along positive direction of $x -$axis it becomes ${P}^{"}\left(b + 2, a\right) .$

Now, applying Rotation theorem

$- \dfrac{1}{\sqrt{2}} + \dfrac{7}{\sqrt{2}}i = \left(\left(b + 2\right) + ai\right)\left(cos\dfrac{\pi}{4} + isin\dfrac{\pi}{4}\right)$

Multiplying the complex numbers we get,

$\dfrac{- 1}{\sqrt{2}} + \dfrac{7}{\sqrt{2}}i = \left(\dfrac{b + 2}{\sqrt{2}} - \dfrac{a}{\sqrt{2}}\right) + i\left(\dfrac{b + 2}{\sqrt{2}} + \dfrac{a}{\sqrt{2}}\right)$

$⇒ b - a + 2 = - 1 . . . \left(1\right)$

and $b + 2 + a = 7 . . . \left(2\right)$

$⇒ a = 4; b = 1$

$⇒ 2a + b = 9