The distance between the vertex and the focus of the parabola $x^2-2x+3y-2=0$ is

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 one of the following curves cuts the parabola $y^2 = 4ax$ at right angles?

What is the equation of the normal to the curve $y^2 = 4ax$ at the point $(a, 2a)$?

For the parabola $y^2 = 8(x - 3)$,let $P$ be a point on it. Let $M$ be the foot of the perpendicular from $P$ to the directrix,and $S$ be the focus of the parabola. If $\triangle SPM$ is an equilateral triangle,find the length of each side of the triangle.

$\triangle OAB$ is an equilateral triangle inscribed in the parabola $y^2 = 4ax, a > 0$ with $O$ as the vertex. Then the length of the side of $\triangle OAB$ is