Let $A(1, 2)$,$B(4, -4)$,and $C(2, 2\sqrt{2})$ be points on the parabola $y^2 = 4x$. If $\alpha$ and $\beta$ respectively represent the area of $\triangle ABC$ and the area of the triangle formed by the tangents at $A, B, C$ to the parabola,then $\alpha \beta =$

The length of the latus rectum of the parabola $4y^{2} + 3x + 3y + 1 = 0$ is

The locus of the point of intersection of the tangents drawn at the ends of a focal chord of the parabola $x^{2}=-8y$ is

If the perpendicular distance from the focus of a parabola $y^2=4ax$ to its directrix is $\frac{3}{2}$,then the equation of the normal drawn at $(4a, -4a)$ is

If $a \ne 0$ and the line $2bx + 3cy + 4d = 0$ passes through the points of intersection of the parabolas $y^2 = 4ax$ and $x^2 = 4ay$,then

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$,such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $R$ denote the region lying in the first quadrant,enclosed by the parabola $y^2=x$,the curve $S$,and the lines $x=1$ and $x=4$. Then which of the following statements is (are) True?
$(A) \ (4, \sqrt{3}) \in S$
$(B) \ (5, \sqrt{2}) \in S$
$(C)$ Area of $R$ is $\frac{14}{3}-2 \sqrt{3}$
$(D)$ Area of $R$ is $\frac{14}{3}-\sqrt{3}$