Let $P$ be the mid-point of a chord joining the vertex of the parabola $y^{2}=8x$ to another point on it. Then,the locus of $P$ 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

$A$ line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan^{-1}\left(\frac{3}{4}\right)$ with the $X$-axis. It intersects the parabola $y^{2}=4(x-3)$ at points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$,respectively. Then $|x_{1}-x_{2}|$ is equal to

The equation of the parabola with the focus $(3,0)$ and the directrix $x+3=0$ is:

The vertex of a parabola is the point $(a, b)$ and the latus rectum is of length $l$. If the axis of the parabola is along the positive direction of the $y$-axis,then its equation is

The coordinates of the focus of the parabola described parametrically by $x=5t^2+2, y=10t+4$ (where $t$ is a parameter) are