Proving Focal chords property of parabola

Question

Consider a parabola $y^2 = 4ax$ from which we draw two focal chords at $t_1, t_2$ respectively . lets say $P1P2$ and $C1C2$ where $C1,C2$ are the other end points intersecting the parabola again . Now my question is it is easy to show that the lines $P1C1$ and $P2C2$ meets at directrix of the parabola? I think this property holds even for ellipse/hyperbola too , but can the proof be such that it may give insight in case of those too ? If this property is valid . Though i have tried using coordinate finding the equations of those lines like one would be $y - 2at_1$ = $\frac{2}{t_1 + t_2} (x-at_1^2)$ but its getting lengthy when solving for intersection point . I hope there is a better geometric approach to it . My coordinate method was this : as above the line joining it is given , the other equation would be $y + \frac{2a}{t_1}$ = $\frac{-2t_1 t_2}{t_1 + t_2} (x-\frac{a}{t_1^2})$ , just using the propety of $t_1 t'_1 = -1$ , where t'_1 is the other point other than t_1 intersecting parabola. Equating the y value we get $x\frac{2(t_1 t_2 + 1)}{t_1 + t_2} = 2a \frac {t_2 - t_1 - t_2 - t^3_1 - t^2_1 t_2 + t^3_1}{(t_1+t_2 )t_1}$ gives x = -a