Statement (I) :The  normal at a point $P$ on the parabola $y^2=4x$  meets the $x-$axis at $Q$. If M is the mid point of the line segment $PQ$, then the locus of M is another parabola whose vertex is at the focus of the given parabola. 
Statement (II) : The tangent at a point $P$ on the parabola $y^2=4x$  meets the $x-$-axis at $Q$. If $R$ is the mid point of $PQ$, then the locus of $R$ is the tangent at the vertex of the parabola. 
 

Normal at the point  $P(m^2,-2m)$ on the parabola $y^2=4xisy=mx-2m-m^3$  which meet the x-axis at  $Q(2+m^2,0).$
Coordinates of M are  $(1+m^2,-m)=(x,y)$
Locus of M is $y^2=x-1$  which is a parabola with vertex (1, 0), the focus of the parabola  $y^2=4x.$ Thus statement-1 is true. 
Next, tangent at a point $P(t^2,2t)$  on the parabola $y^2=4x$  is $ty=x+t^2$  which meets the x-axis at $Q(-t^2,0)$  coordinates of R are (0, t) = (x, y)
Locus of R is x = 0, the tangent at the vertex of parabola $y^2=4x.$  So statement – II is also true but does not lead to statement - I