If $A_1{B}_1\text{ and }{A}_2{B}_2$  are two focal chords of the parabola $y^2=4ax\text{ then the chords }{A}_1{A}_2\text{ and }{B}_1{B}_2$  intersect on 

$A_1{B}_1\text{ is a focal chord, then if }{A}_1\text{ be }{t}_1\text{ then }{B}_1\text{ is }-1/t_1\text{ : }$

Similarly $A_2\text{ is }{t}_2\text{ and }{B}_2\text{ is }-1/t_2\text{. }$

Chord $A_1{A}_2\text{ is }y\left(t_1+t_2\right)-2x-2a{t}_1{t}_2=0$     ... (1)

Chord $B_1{B}_2$  is 

       $y\left(-\frac{1}{t_1}-\frac{1}{t_2}\right)-2x-2a\left(-\frac{1}{t_1}\right)\left(-\frac{1}{t_2}\right)=0$

$\Rightarrow$ $y\left(t_1+t_2\right)+2x{t}_1{t}_2+2a=0$      ... (2) 

For their intersection we subtract them and get 

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2x\left(t_1{t}_2+1\right)+2a\left(t_1{t}_2+1\right)=0\\ \text{ ⇒}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(x+a)\left(1+t_1{t}_2\right)=0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x+a=0\text{ i.e., directrix. }\end{array}$