The condition that the line $y=mx+c$ to be a tangent to the parabola $y^2=4a(x+a)$ is

$y=mx+c$,$y^2=4a(x+a)$

$\Rightarrow {(mx+c)}^2=4ax+4a^2\Rightarrow m^2{x}^2+c^2+2mcx=4ax+4a^2$

$\Rightarrow m^2{x}^2+2x(mc-2a)+(c^2-4a^2)=0$

Discriminant is zero

$\Rightarrow 4{(mc-2a)}^2=4m^2(c^2-4a^2)$

$c=a(m+\frac{1}{m})$