If the vertex of a parabola is $(2, -1)$ and the equation of its directrix is $4x - 3y = 21$,then the length of its latus rectum is

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x = -3$ are in the ratio $3:1$. If $R(\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$,then $\frac{\beta^2}{\alpha}$ is equal to $.............$.

If the parabola $y^2 = 4ax$ passes through the point $(-3, 2)$,then the length of its latus rectum is

Find the equation of the tangent to the parabola $x^2 = y$ at one of the endpoints of its latus rectum in the first quadrant.

Let $A(0,1)$,$B(1,1)$,and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y^2 = 4ax$ passing through $D$ is $(\alpha + \beta \sqrt{2}, 0)$,where $\alpha$ and $\beta$ are rational numbers,then $\frac{\alpha}{\beta^2}$ is equal to

If $P$ is a point on the parabola $y^2=8x$ and $A$ is the point $(1,0)$,then the locus of the mid-point of the line segment $AP$ is