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 $x-y-3=0$ is a normal drawn through the point $(5,2)$ to the parabola $y^2=4x$,then the slope of the other normal that can be drawn through the same point to the parabola $y^2=4x$ is

The locus of the mid-point of the line segment joining the focus and any point on the parabola $y^{2}=4ax$ is a parabola. Find the equation of its directrix.

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15x$ subtends an angle $\theta$ at the vertex of the parabola,then $\sin \frac{\theta}{3}+\cos \frac{2\theta}{3}-\sec \frac{4\theta}{3}=$

The area of a triangle (in sq. units) formed by the latus rectum of the parabola $x^2=16y$ and the lines joining the vertex of the parabola to the ends of the latus rectum is

Let $y^{2}=12x$ be the parabola with its vertex at $O(0,0)$. Let $P$ be a point on the parabola and $A$ be a point on the $x$-axis such that $\angle OPA=90^{\circ}$. Then the locus of the centroid of such triangles $OPA$ is: