The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4x$ is

The curve described parametrically by $x = t^2 - 2t + 2$ and $y = t^2 + 2t + 2$ represents:

If the vertex of a parabola is $(0, a)$ and the focus is $(0, 0)$,what is its equation?

The equation of the latus rectum of a parabola is $x + y = 8$ and the equation of the tangent at the vertex is $x + y = 12$. The length of the latus rectum 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 $2$.

The locus of the middle points of the chords of the parabola $y^2 = 4ax$ which pass through the origin is: