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

Let $A$ and $B$ be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $d$ denotes the distance between $A$ and $B$,and $a$ denotes the area of $\triangle SAB$,where $S$ is the focus of the parabola $y^2=4x$,then the value of $(a+d)$ is:

For the parabola $y^2+6y-2x+5=0$,match the items in List-$I$ with the suitable item in List-$II$ given below:

List-$I$ (Geometric Property)	List-$II$ (Coordinates/Equations)
$I$. Vertex	$A$. $\left(-\frac{3}{2}, -3\right)$
$II$. Focus	$B$. $\left(\frac{3}{2}, -3\right)$
$III$. Equation of the directrix	$C$. $2x + 5 = 0$
$IV$. Equation of the axis	$D$. $2x + y + 3 = 0$
	$E$. $y + 3 = 0$
	$F$. $(-2, -3)$

The correct matching is:

Two tangent lines $l_{1}$ and $l_{2}$ are drawn from the point $(2,0)$ to the parabola $2y^{2} = -x$. If the lines $l_{1}$ and $l_{2}$ are also tangent to the circle $(x-5)^{2} + y^{2} = r$,then $17r$ is equal to.

Let $S$ be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b, c)$,where $b, c \in \mathbb{N}$,on the parabola $y^2 = 2ax$ and the lines $x = b$ and $y = 0$ is $16 \text{ unit}^2$. Then $\sum_{a \in S} a$ is equal to $..........$.

The equation of the directrix of the parabola $y^2-x+4y+5=0$ is