The equation of the tangent to the parabola $y^2 = 4ax$ at the point $(3, 2)$ is . . . .

If two tangents drawn from a point $P$ to the parabola $y^2 = 4x$ are such that the slope of one tangent is double the other,then $P$ lies on the curve:

The ends of the latus rectum of the conic ${x^2} + 10x - 16y + 25 = 0$ are

Let $P(x, y)$ be the midpoint of the line joining $(1, 0)$ to a point on the curve $y^{2} = \left|\begin{array}{ll}x+1 & x+2 \\ x+3 & x+5\end{array}\right|$. Then,the locus of $P$ is symmetrical about

The diameter of the parabola $y^2 = 4ax$ which bisects the chords parallel to $y = mx + \alpha$ 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: