Find the angle $\alpha$ between the tangents drawn from the point $(-2, -1)$ to the parabola $y^2 = 4x$. What is the value of $\tan \alpha$?

$A$ circle is drawn with its centre at the focus of the parabola $y^2 = 2px$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is

Tangents are drawn from the point $(-1, 2)$ to the parabola $y^2 = 4x$. The length of the intercept made by these tangents on the line $x = 2$ is:

The equation of the line touching both parabolas $y^2=4x$ and $x^2=-32y$ is

The straight line joining any point $P$ on the parabola $y^2 = 4ax$ to the vertex and the perpendicular from the focus to the tangent at $P$ intersect at $R$. Then the equation of the locus of $R$ is:

If the equation of a system of parallel chords of the parabola $y^2 = \frac{2}{3}x$ is $y + 2x + 1 = 0$,find its diameter.