Parabolas $y^2=4a\left(x–k\right)$ and $x^2=4a\left(y–k\right)$ touche each other at line $y=x$ 
($\because$ both parabola are inverse of each other)
$\Rightarrow$ $y=x$ is a common tangent at $P$
$\Rightarrow$ we get point $P$ by solving $y^2=4a\left(x–k\right)$ and $y=x$
$\Rightarrow$ point of contact $P$ is $\left(2a,2a\right)$
$\Rightarrow$ equation of normal at $P\left(2a,2a\right)$ is $x+y=4a$