The equation of the tangent to the parabola $y^2=8x$,which is parallel to the line $4x-y+3=0$,is

If the tangent to the curve $x = at^2, y = 2at$ is perpendicular to the $x$-axis,then what is the point of contact?

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:

Find the locus of the midpoint of the chord of the parabola $y^2 = 4x$ drawn from the vertex.

If $PSQ$ is a focal chord of the parabola $y^2 = 8x$ such that $SP = 6$,then find the length of $SQ$.

Let $A, B$ and $C$ be the vertices of a variable right-angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex containing the right angle be $C = (4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of the latus rectum of $C_o$ is . . . . . .