What is the equation of the normal to the parabola $y^2 = 4ax$ at the point $(\frac{a}{m^2}, \frac{2a}{m})$?

If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12x$ and a point on the parabola in the ratio $1:2$,then the locus of $P$ is:

What is the distance between the focus and the directrix of the parabola $x^2 = -8y$?

The equation of the parabola with focus $(1, -1)$ and directrix $x+y+3=0$ is

Let $P(x, y)$ be the midpoint of the line joining $(1, 0)$ to a point on the curve $y^{2} = \left|\begin{array}{ll}x+1 & x+2 \\ x+3 & x+5\end{array}\right|$. Then,the locus of $P$ is symmetrical about

If the line $y - \sqrt{3}x + 3 = 0$ cuts the parabola $y^2 = -x - 2$ at $A$ and $B$,then $PA \cdot PB$ is equal to,where $P \equiv (\sqrt{3}, 0)$.