$P$is a point on the locus of the mid-points of the
chords of the parabola $y^2=4ax$ passing through
the vertex of the parabola; $S$is the focus of the
parabola and the line joining $S$and $P$is produced

to meet the parabola at $Q$. If the ordinate of $P$is
equal to its abscissa, coordinates of $Q$are

$O(0, 0)$ is the vertex and $R\left(a{t}^2,2at\right)$ is any point
on the parabola. $\left(a{t}^2/2,2at/2\right)$ is the mid-point of
the chord $OR$. Whose locus is $y^2=2ax$ coordinates
of P on this locus are $(2a, 2a)$, focus is 

$S(a, 0).$ Equation of $SP$ is $y=2(x-a)$ which meets

$y^2=4ax$at the point $Q$for which $4(x-a{)}^2=4ax$

$\Rightarrow x^2-3ax+a^2=0$ which is not satisfied by the
points in$\left(a\right), \left(b\right) or \left(c\right).$