Show that the condition that the line y=mx+c may be a tangent to the parabola y² = 4ax is $\mathrm{c}=\frac{\mathrm{a}}{\mathrm{m}}$

Let P (x₁, y₁) be the point where the line y = mx + c touches the parabola y² = 4ax.
The equation of the tangent at (x₁, y₁) on the parabola is ${\mathrm{yy}}_1-2\mathrm{a}\left(\mathrm{x}+{\mathrm{x}}_1\right)=0 ...\left(1\right)$
Given tangent is y - mx - c = 0 ... (2)
(1) and (2) represent the same line
$\Rightarrow \frac{{\mathrm{y}}_1}{1}=\frac{2\mathrm{a}}{\mathrm{m}}=\frac{2{\mathrm{ax}}_1}{\mathrm{c}}\Rightarrow {\mathrm{x}}_1=\frac{\mathrm{c}}{\mathrm{m}}\text{ and }{\mathrm{y}}_1=\frac{2\mathrm{a}}{\mathrm{m}}$
Since P(x₁, y₁) lies on the parabola then
$$\begin{gathered} {\mathrm{y}}_1^2=4{\mathrm{ax}}_1 \\ \Rightarrow \frac{4{\mathrm{a}}^2}{{\mathrm{m}}^2}=4\mathrm{a}\left(\frac{\mathrm{c}}{\mathrm{m}}\right) \\ \Rightarrow \mathrm{c}=\frac{\mathrm{a}}{\mathrm{m}} \end{gathered}$$