Explanation

(d)
Since, The intersection of a line $y = 2x + c$ and a parabola $y^{2} = 4ax + 4a^{2}$ is
$(2x + c)^{2} = 4ax + 4a^{2}$
$\Rightarrow 4x^{2} + 4(c - a)x + \left( c^{2} - 4a^{2} \right) = 0$
Since, it is a tangent line
$\therefore 16(c - a)^{2} - 4 \times 4\left( c^{2} - 4a^{2} \right) = 0\ \ \ \lbrack\therefore D = 0\rbrack$
$\Rightarrow \ c^{2} + a^{2} - 2ac - c^{2} + 4a^{2} = 0$
$\Rightarrow \ \ c = \frac{5a}{2}$