If a focal chord of the parabola $y^2 = 4ax$ makes an angle $\theta$ with its axis,then the length of the perpendicular from the vertex to this chord is......

The eccentricity of the parabola $x^2 - 4x - 4y + 4 = 0$ is

Find the directrix of the locus of the midpoint of the line segment joining the focus and a variable point on the parabola $y^{2} = 4ax$.

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 $2$.

Find the equation of the parabola with focus $(2, 0)$ and directrix $x = -2$.

The equation of the tangent at $P(-4, -4)$ on the curve $x^{2} = -4y$ is