The directrix of the parabola ${x^2 - 4x - 8y + 12 = 0}$ is

The largest value of $k$ for which the circle $x^2+y^2=k^2$ lies completely in the interior of the parabola $y^2=4x+16$ is

For the parabola $y^2 = 4ax$,what is the $x$-coordinate of the point closest to the focus?

What is the equation of the pair of tangents drawn from the point $(1, 4)$ to the parabola $y^2 = 12x$?

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y =-\frac{1}{2}$. Then $S=\left\{x \in R : \tan ^{-1}\left(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1})\right)=\frac{\pi}{2}\right\}$:

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the endpoints of its latus rectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola,and $\Delta_2$ be the area of the triangle formed by the intersection points of the tangents drawn at $P$ and at the endpoints of the latus rectum. Then $\frac{\Delta_1}{\Delta_2}$ is