Find the condition for the line y =mx+c to be a tangent to the parabola x²=4ay

Let (x₁, y₁) be the point of contact
The equation of tangent is ${\mathrm{xx}}_1=2\mathrm{a}\left(\mathrm{y}+{\mathrm{y}}_1\right)$
$\Rightarrow {\mathrm{xx}}_1-2\mathrm{ay}-2{\mathrm{ay}}_1=0.....\left(1\right)$
Comparing the coefficients of (1) and mx-y+c = 0
$\Rightarrow \frac{{\mathrm{x}}_1}{\mathrm{m}}=\frac{-2\mathrm{a}}{-1}=\frac{-2{\mathrm{ay}}_1}{\mathrm{c}}\Rightarrow {\mathrm{x}}_1=2\mathrm{am};{\mathrm{y}}_1=-\mathrm{c}$
Substituting (2am, -c) on the parabola x² =4ay
$\Rightarrow \frac{{\mathrm{x}}_1}{\mathrm{m}}=\frac{-2\mathrm{a}}{-1}=\frac{-2{\mathrm{ay}}_1}{\mathrm{c}}\Rightarrow {\mathrm{x}}_1=2\mathrm{am};{\mathrm{y}}_1=-\mathrm{c}$
Substituting (2am, -c) on the parabola x² =4ay
$4{\mathrm{a}}^2{\mathrm{m}}^2=4\mathrm{a}(-\mathrm{c})\Rightarrow \mathrm{c}=-{\mathrm{am}}^2$
$\therefore$The condition is $\mathrm{c}=-{\mathrm{am}}^2$