Tangents at the extremities of a focal chord of a
parabola intersect

Extremities of a focal chord are $\left(a{t}^2,2at\right),\left(a/t^2,-2a/t\right)$

Tangents are $ty=x+a{t}^2,\text{ and }-ty=x{t}^2+a$

They intersect at $\left(-a,\frac{a\left(t^2-1\right)}{t}\right)\text{ or }\left(-a,a\left(t-\frac{1}{t}\right)\right)$

which is the point of intersection of the directrix

$x = – a$ and the line $y=a\left(t-\frac{1}{t}\right)$ which is the
line through the mid-point of the chord parallel to
the axis.