Find the vertex of the parabola $y^2 + 6x - 2y + 13 = 0$.

Let $M$ be the foot of the perpendicular from a point $P$ on the parabola $y^2=8(x-3)$ onto its directrix and let $S$ be the focus of the parabola. If $\triangle SPM$ is an equilateral triangle,then $P$ is equal to

The maximum number of common normals of $y^2=4ax$ and $x^2=4by$ is equal to

If $PQ$ is a focal chord of the parabola $y^2=4x$ with focus $S$ and $P=(4,4)$,then $SQ=$

Let $P(x, y)$ be a variable point on the parabola $y = 4x^2 + 1$. Let $Q(c, c)$ be the foot of the perpendicular drawn from $P$ to the line $y = x$. If $R(h, k)$ is the mid-point of $PQ$,then the locus of $R$ is:

The locus of a point which divides the line segment joining the focus and any point on the parabola $y^2 = 12x$ in the ratio $m:n$ $(m+n \neq 0)$ is a parabola. Then the length of the latus rectum of that parabola is