If $PSQ$ is a focal chord of the parabola $y^2 = 8x$ such that $SP = 6$,then find the length of $SQ$.

Let $x=2t, y=\frac{t^2}{3}$ be a conic. Let $S$ be the focus and $B$ be the point $(0, \alpha)$ on the axis of the conic such that $SA \perp BA$,where $A$ is any point $(2t, \frac{t^2}{3})$ on the conic. If $k$ is the ordinate of the centroid of $\Delta SAB$,then $\lim_{t \rightarrow 1} k$ is equal to

From the point $(-1, -6)$,two tangents are drawn to the parabola $y^2 = 4x$. The angle between the two tangents is:

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

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

If the line $2x - 3y = k$ is tangent to the parabola $y^2 = 6x$,then what is the value of $k$?