Find the equation of the parabola that satisfies the following conditions: Focus $(0, -3)$,directrix $y = 3$.

The equations of the tangents to the parabola $y^2 = 4ax$ at the ends of its latus rectum are-

$A$ triangle $ABC$ with area $5a^{2}$ is inscribed in the parabola $y^{2} = 4ax$,where the vertex $A$ is at the vertex of the parabola and $BC$ is a focal chord. Find the length of the focal chord.

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?

The equation of the parabola whose focus is the point $(0, 0)$ and the tangent at the vertex is $x - y + 1 = 0$ is

If the axis of a parabola is horizontal and it passes through the points $(0, 0), (0, -1)$ and $(6, 1)$,then its equation is