Let $P_1 : y = -x^2 + 4x + 2$ and $P_2 : x^2 + 5x + \frac{17}{8} = y$ be two parabolas. Then,the number of common tangents of $P_1$ and $P_2$ is:

$A$ ray of light moving parallel to the $x$-axis gets reflected from a parabolic mirror whose equation is $(y - 2)^2 = 4(x + 1)$. After reflection,the ray must pass through the point:

The point on the parabola $2y = x^2$ which is nearest to the point $(0, 3)$ is

The distance of the point $(6, -2 \sqrt{2})$ from the common tangent $y = mx + c$ $(m > 0)$ of the curves $x = 2y^2$ and $x = 1 + y^2$ is

If the line $y = -x + k$ is normal to the curve $y^2 = 16x$,then $k$ is

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $x^{2}=-9y$.