Equation of the tangent at the point P (t), where t is any parameter, to the parabola y² = 4ax is

$y t=x+a t^2$

Coordinates of the point P are (at², 2at)

Differentiating $y^2=4 a x \Rightarrow 2 y \frac{d y}{d x}=4 a$

$\Rightarrow \frac{d y}{d x}=\left.\frac{2 a}{y} \Rightarrow \frac{d y}{d x}\right|_P=\frac{2 a}{2 a t}=\frac{1}{t}$

∴ equation of tangent is

$y-2 a t=\frac{1}{t}\left(x-a t^2\right) \Rightarrow x-y t+a t^2=0$