Plz note maxzy focal chord and latus rectum are not same.A chord to the parabola passing thro' focus is called a focal chord.

Now,let us assume the vertex of the parabola as origin and its axis as x-axis.Then it's eqn. becomes y²=4ax with focus (a,0) and directrix x+a=0.

Let P(t),Q(t') be the extremities of the focal chord then tt'=-1.So,P=(at²,2at) and Q=(a/t²,-2a/t).

PQ²=a²(t+1/t)⁴PQ=a(t+1/t)² and so,radius R of the circle=(a/2)(t+1/t)².

Now,the circle drawn with PQ as diameter touches the directrix iff

Distance d of the directrix is equal to radius R i.e;d=R.

Here,Centre of circle(x₀,y₀)=((a/2)(t²+1/t²),a(t-1/t))

and so,d=Ix₀+aI=(a/2)(t²+1/t²)+a=(a/2)[t²+1/t²+2]=(a/2)(t+1/t)²=R.

Hence,the directrix touches the given circle.