The tangent at the end points of the focal chord meets at the directrix.

The equation of the tangent at $P$ is $y\cdot 2=-\frac{\left(x-4\right)}{2}$
$\Rightarrow 4y=-x+4$

This tangent intersects the directrix at $\left(\frac{1}{4},\frac{15}{16}\right)$