If $2x + y + \lambda = 0$ is a focal chord of the parabola $y^2 = -8x$,find the value of $\lambda$.

Consider the parabola $25[(x-2)^2+(y+5)^2]=(3x+4y-1)^2$. Match the characteristics of this parabola given in List-$I$ with their corresponding items in List-$II$.

List-$I$	List-$II$
$I$. Vertex	$A$. $8$
$II$. Length of latus rectum	$B$. $(\frac{29}{10}, \frac{-38}{10})$
$III$. Directrix	$C$. $3x+4y-1=0$
$IV$. One end of the latus rectum	$D$. $(\frac{-2}{5}, \frac{-16}{5})$
	$E$. $6$

Let one end of a focal chord of the parabola $y^{2}=16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5 : 2$, then the minimum value of $\alpha+\beta$ is equal to:

Find the equation of the normal to the parabola $y^2 + 12x = 0$ at the upper end of its latus rectum.

Let the tangent and normal at any point $P(at^2, 2at)$, $(a > 0)$, on the parabola $y^2 = 4ax$ meet the axis of the parabola at $T$ and $G$ respectively. Then the radius of the circle through $P, T$ and $G$ is

Find the equation of the common tangent to the parabolas $y = x^2$ and $y = -(x - 2)^2$.