Two parabolas $y^2 = 4a(x - l_1)$ and $x^2 = 4a(y - l_2)$ always touch one another,where $l_1$ and $l_2$ are variable. The locus of their point of contact has the equation:

The locus of the vertices of the family of parabolas $6y = 2a^3x^2 + 3a^2x - 12a$ is

If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12x$ and a point on the parabola in the ratio $1:2$,then the locus of $P$ is:

If the focal chord of the parabola $x^2=12y$ drawn through the point $(3,0)$ intersects the parabola at the points $P$ and $Q$,then the sum of the reciprocals of the abscissae of the points $P$ and $Q$ is

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 $\frac{14}{3}-\sqrt{3}$

The equation of a straight line drawn through the focus of the parabola $y^2 = -4x$ at an angle of $120^\circ$ to the $x$-axis is: