The locus of the points of intersection of perpendicular normals to the parabola $y^2=4ax$ is

Let $a, r, s, t$ be nonzero real numbers. Let $P(at^2, 2at)$,$Q(at'^2, 2at')$,$R(ar^2, 2ar)$,and $S(as^2, 2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel,where $K$ is the point $(2a, 0)$.
$1.$ The value of $r$ is
$(A) -\frac{1}{t}$ $(B) \frac{t^2+1}{t}$ $(C) \frac{1}{t}$ $(D) \frac{t^2-1}{t}$
$2.$ If $st=1$,then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is
$(A) \frac{(t^2+1)^2}{2t^3}$ $(B) \frac{a(t^2+1)^2}{2t^3}$ $(C) \frac{a(t^2+1)^2}{t^3}$ $(D) \frac{a(t^2+2)^2}{t^3}$
Give the answer for question $1$ and $2$.

Let $O$ be the origin and $A$ be a point on the curve $y^2=4x$. Then the locus of the midpoint of $OA$ is:

The line $y = 6x + 1$ touches the parabola $y^2 = 24x$. The coordinates of a point $P$ on this line,from which the tangent to $y^2 = 24x$ is perpendicular to the line $y = 6x + 1$,is

What is the equation of the directrix of the parabola $x^2 = -8y$?

If the three normals drawn to the parabola $y^{2} = 2x$ pass through the point $(a, 0)$ where $a \neq 0$,then $a$ must be greater than: