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:

If one end of a focal chord of the parabola $y^2=8x$ is $\left(\frac{1}{2}, 2\right)$,then the length of the focal chord is $........$ units.

Suppose $AB$ is a focal chord of the parabola $y^2=12x$ of length $l$ and slope $m < \sqrt{3}$. If the distance of the chord $AB$ from the origin is $d$,then $l \cdot d^2$ is equal to ....................

If a point $P$ moves such that its distances from the point $A(1, 1)$ and the line $x+y+2=0$ are equal,then the locus of $P$ is

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$,then the distance between $A$ and $B$ is

The equation of the parabola with its vertex at the origin,axis on the $y$-axis and passing through the point $(6, -3)$ is