Equation of a parabola is
$y^2 = 4ax$
on differentiating w.r.t x we get,
$2y\frac{dy}{dx} = 4a$
$\frac{dy}{dx} = \frac{2a}{y}$
$\frac{dy}{dx}$ at $(a{t_1}^2,2a{t_1}) = \frac{2a}{2a{t_1}} = \frac{1}{{t_1}}$
Hence $m = \frac{1}{{t_1}}$
Equation of a tangent is $y-{y_1}=m(x-{x_1})$
$y-2a{t_1} = \frac{1}{t1}(x-a{t_1}^2)$
on simplifying we get
$y{t_1} -2a{t_1}^2 = x-a{t_1}^2$
that is $y{t_1} = x + a{t_1}^2$
Hence proved.
Easy way to remember the equation of any tangent to a curve is $y-{y_1}=m(x-{x_1})$