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

Find the length of the chord of the parabola $y^2 = 4x$ which passes through the vertex and makes an angle of $30^{\circ}$ with the $x$-axis. (in $\sqrt{3}$)

Let chord $PQ$ of length $3\sqrt{13}$ of the parabola $y^2 = 12x$ be such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$. If the chord $PQ$ subtends an angle $\alpha$ at the focus of the parabola, then $\sin \alpha$ is equal to:

If the vertex and focus of a parabola lying on the $x$-axis are at distances $p$ and $q$ from the origin respectively,find its equation.

Variable chords of the parabola $y^2 = 4ax$ subtend a right angle at the vertex. Then:

Find the equation of the parabola whose axis is parallel to the $y$-axis and which passes through the points $(0,4), (1,9)$ and $(4,5)$.