if $x=my+c$ is a normal to the parabola $x^2=4ay$,  then the value of c is 

$y=mx-2am-a{m}^3$ is normal to parabola $y^2=4ax$

$\therefore x=my-2am-a{m}^3$ is normal to parabola $x^2=4ay$ by interchanging x, y 

Comparing with $x=my+c,c=-2am-a{m}^3$