If the line $2x + y + k = 0$ is normal to the parabola $y^2 = -8x$,then the value of $k$ will be

The number of common tangents to the parabolas $y = x^{2}$ and $y = -x^{2} + 4x - 4$ is:

The equation of the parabola whose vertex is at $(2, -1)$ and focus at $(2, -3)$ is

The vertex of the parabola $(y - 2)^2 = 16(x - 1)$ is

The coordinates of the ends of a focal chord of a parabola $y^2 = 4ax$ are $(x_1, y_1)$ and $(x_2, y_2)$. Then $x_1x_2 + y_1y_2$ has the value equal to

$A$ quadratic polynomial $y = f(x)$ with absolute term $3$ neither touches nor intersects the abscissa axis and is symmetric about the line $x = 1$. The coefficient of the leading term of the polynomial is unity. $A$ point $A(x_1, y_1)$ with abscissa $x_1 = 1$ and a point $B(x_2, y_2)$ with ordinate $y_2 = 11$ are given in a Cartesian rectangular system of coordinates $OXY$ in the first quadrant on the curve $y = f(x)$,where $O$ is the origin. The graph of $y = f(x)$ represents a parabola whose focus has the coordinates: