Let $y^2 = 12x$ be the parabola with its vertex at $O$. 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:

Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16,16)$. If $P(\alpha,\ \beta)$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha + \beta$ is equal to:

Let O be the vertex of the parabola $x^2=4 y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio $2: 3$ be the conic C . Then the equation of the chord of $C$, which is bisected at the point $(1,2)$, is :

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P, whose abscissa is $-$2, is