If the parabola $x^{2} = 4ay$ passes through the point $(2, 1)$,then the length of the latus rectum is

Find the length of the chord of the parabola $x^2 = 4ay$ which passes through the vertex and has a slope of $\tan \alpha$.

Two perpendicular tangents to the parabola $y^2 = 4ax$ always intersect on the line:

The equation of the parabola with its vertex at $(1, 1)$ and focus at $(3, 1)$ is

If the line segment joining the vertex of the parabola $y^2=4ax$ and a point on the parabola makes an angle $\theta$ with the positive $X$-axis,then the length of that line segment is

Let the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2 + q^2$ is: