The parametric equations of the parabola $y^2-4x-8y-12=0$ are

The length of the chord of the parabola $y^2 = 4ax$ which passes through the vertex and makes an angle $\theta$ with the axis of the parabola is:

Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $P(1,3)$ on it meet the $y$-axis at $A$. Let the line passing through $P$ and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at $B$. If $B$ lies on the line $2x-3y=8$,then $(AB)^2$ is equal to $............$.

From the point $(-1, 2)$, tangents are drawn to the parabola $y^2 = 4x$. The area of the triangle formed by the chord of contact and the tangents is: (in $\sqrt{2}$)

If $(2, -8)$ is one endpoint of a focal chord of the parabola $y^{2} = 32x$,what is the other endpoint?

The equation of the directrix of the parabola $y^2-x+4y+5=0$ is