If the line $x = my + k$ touches the parabola $x^2 = 4ay$,then $k = $

Tangents drawn at the ends of any focal chord of a parabola $y^2 = 4ax$ intersect on the line

Two parabolas with a common vertex at the origin and with axes along the $x-$ axis and $y-$ axis,respectively,intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$,then the equation of the common tangent to the two parabolas is?

Suppose a parabola $y=ax^2+bx+c$ has two $x$-intercepts,one positive and one negative,and its vertex is $(2,-2)$. Then,which of the following is true?

Let $P(\alpha, \beta)$ be a point on the parabola $y^2 = 4x$ which is at minimum distance from the circle $x^2 + y^2 - 4x - 20y + 103 = 0$. Then $\alpha \beta$ is

Let $P_{1}$ be a parabola with vertex $(3,2)$ and focus $(4,4)$,and let $P_{2}$ be its mirror image with respect to the line $x + 2y = 6$. Then the directrix of $P_{2}$ is $x + 2y =$