Find the equation of a straight line joining the pair of points $\left(at_1^2, 2at_1\right)$ and $\left(at_2^2, 2at_2\right)$.
Let $\;$ $A \left(x_1, y_1\right) = \left(at_1^2, 2at_1\right)$, $\;$ $B \left(x_2, y_2\right) = \left(at_2^2, 2at_2\right)$
Slope of the required line $= m = \dfrac{y_2 - y_1}{x_2 - x_1}$
i.e. $\;$ $m = \dfrac{2at_2 - at_1}{at_2^2 -at_1^2}$
i.e. $\;$ $m = \dfrac{2a \left(t_2 - t_1\right)}{a \left(t_2^2 - t_1^2\right)}$
i.e. $\;$ $m = \dfrac{2 \left(t_2 - t_1\right)}{\left(t_2 - t_1\right) \left(t_2 + t_1\right)}$
i.e. $\;$ $m = \dfrac{2}{t_1 + t_2}$
Equation of the required line is of the form: $\;$ $y - y_1 = m \left(x - x_1\right)$
i.e. $\;$ $y - 2at_1 = \left(\dfrac{2}{t_1 + t_2}\right) \left(x - at_1^2\right)$
i.e. $\;$ $y \left(t_1 + t_2\right) - 2 a t_1 \left(t_1 + t_2\right) = 2x - 2a t_1^2$
i.e. $\;$ $2x - y \left(t_1 + t_2\right) + 2at_1 \left(t_1 + t_2\right) - 2 a t_1^2 = 0$
i.e. $\;$ $2x - y \left(t_1 + t_2\right) + 2 a t_1 \left(t_1 + t_2 - t_1\right) = 0$
i.e. $\;$ $2x - y \left(t_1 + t_2\right) + 2 a t_1 t_2 = 0$