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If $P_1 P_2$ and $P_3 P_4$ are two focal chords of the parabola $y^2=4 a x$ then the chords $P_1 P_3$ and $P_2 P_4$ intersect on the
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directrix of the parabola
Let $P_i\left(a_i{ }^2, 2 a t_i\right), i=1,2,3,4$
then, $t_1 t_2=t_3 t_4=-1$
equation of $P_1 P_3$ : $\left(t_1+t_3\right) y=2 x+2 a t_1 t_3$ ...(1)
$P_2 P_4:\left(t_2+t_4\right) y=2 x+2 a t_2 t_4$ ...(2)
Putting $x=-a$ in equation (1) gives $y=\frac{2 a t_1 t_3-2 a}{t_1+t_3}$ Putting $x=-a$ in equation (2) gives
$y=\frac{2 \mathrm{at}_2 \mathrm{t}_4-2 \mathrm{a}}{\mathrm{t}_2+\mathrm{t}_4}=\frac{\frac{2 \mathrm{a}}{\mathrm{t}_4 \mathrm{t}_3}-2 \mathrm{a}}{-\frac{1}{\mathrm{t}_1}-\frac{1}{\mathrm{t}_3}}=\frac{2 \mathrm{a}\left(1-\mathrm{t}_1 \mathrm{t}_3\right)}{-\left(\mathrm{t}_1+\mathrm{t}_3\right)}=\frac{2 \mathrm{a}\left(\mathrm{t}_1 \mathrm{t}_3-1\right)}{\left(\mathrm{t}_1+\mathrm{t}_3\right)}$
$\therefore$ These lines meet at $\left(-\mathrm{a}, \frac{2 \mathrm{a}\left(\mathrm{t}_1 \mathrm{t}_3-1\right)}{\mathrm{t}_1+\mathrm{t}_3}\right)$
then, $t_1 t_2=t_3 t_4=-1$
equation of $P_1 P_3$ : $\left(t_1+t_3\right) y=2 x+2 a t_1 t_3$ ...(1)
$P_2 P_4:\left(t_2+t_4\right) y=2 x+2 a t_2 t_4$ ...(2)
Putting $x=-a$ in equation (1) gives $y=\frac{2 a t_1 t_3-2 a}{t_1+t_3}$ Putting $x=-a$ in equation (2) gives
$y=\frac{2 \mathrm{at}_2 \mathrm{t}_4-2 \mathrm{a}}{\mathrm{t}_2+\mathrm{t}_4}=\frac{\frac{2 \mathrm{a}}{\mathrm{t}_4 \mathrm{t}_3}-2 \mathrm{a}}{-\frac{1}{\mathrm{t}_1}-\frac{1}{\mathrm{t}_3}}=\frac{2 \mathrm{a}\left(1-\mathrm{t}_1 \mathrm{t}_3\right)}{-\left(\mathrm{t}_1+\mathrm{t}_3\right)}=\frac{2 \mathrm{a}\left(\mathrm{t}_1 \mathrm{t}_3-1\right)}{\left(\mathrm{t}_1+\mathrm{t}_3\right)}$
$\therefore$ These lines meet at $\left(-\mathrm{a}, \frac{2 \mathrm{a}\left(\mathrm{t}_1 \mathrm{t}_3-1\right)}{\mathrm{t}_1+\mathrm{t}_3}\right)$
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