If $x+y=k$ is a normal to the parabola $y^2 = 12x$, then it touches the parabola 

Since $y=mx-2am-a{m}^3$ is a normal to the parabola 

$y^2=4ax,$taking $a=3$ and $m=-1$

$y=-x-2\left(3\right)(-1)-3(-1{)}^3$ 

$\Rightarrow x+y=9$ is a normal to the parabola $y^2=12x$. 

Suppose it touches the parabola $y^2=4ax$
Equation of a tangent to the parabola$y^2=4ax$ is 

$y=mx+a/m$ 

If it represents the line  $x+y=9$ then 

$m=-1$ and $a/m=9\Rightarrow a=-9$ 

So an equation of the required parabola is $y^2=4(-9)x$ $\Rightarrow$ 

$y^2=-36x$