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 $\sqrt{3}$)

Find the length of the latus rectum of the parabola whose focus is $(2, 3)$ and the directrix is the line $x - 4y + 3 = 0$.

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

Let $O$ be the vertex of the parabola $y^2 = 4x$ and its chords $OP$ and $OQ$ are perpendicular to each other. If the locus of the mid-point of the line segment $PQ$ is a conic $C$, then the length of its latus rectum is:

Consider the parabola $y^{2}=4x$. Let $P$ and $Q$ be points on the parabola where $P(4, -4)$ and $Q(9, 6)$. Let $R$ be a point on the arc of the parabola between $P$ and $Q$. Then, the area of $\Delta PQR$ is largest when

The equation of the parabola with the focus $(3,0)$ and the directrix $x+3=0$ is: