What is the point of intersection of the latus rectum and the axis of the parabola $y^2 - 4y - 2x - 8 = 0$?

Let the line $y - \sqrt{3}x + 3 = 0$ cut the parabola $2y^2 = 2x + 3$ at $A$ and $B$. If $P(\sqrt{3}, 0)$,then the value of $|PA - PB|$ is [where $PA$ denotes the distance between points $P$ and $A$].

If two circles $x^2+y^2-6x-6y+13=0$ and $x^2+y^2-8y+9=0$ intersect at $A$ and $B$,then the focus of the parabola whose directrix is the line $AB$ and vertex is the point $O(0,0)$ is

Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,passing through $(2, 3)$,and the axis is along the $x$-axis.

The locus of the point $P=(a, b)$ where $a, b$ are real numbers such that the roots of $x^3+a x^2+b x+a=0$ are in arithmetic progression is

Which of the following represents a parabola?