What are the properties of center of curvature of a parabolic mirror? What is so special about the center of curvature of a parabola (since a parabola can't actually have a center) ? I mean, it must be something unique for it to be defined. Why is it twice the focal length? What are the geometrical properties of the center of curvature, in a mathematical sense (like with focus, coefficient of x^2=1/4p) ?
 A: A parabola is a conic section, like circles and ellipses, and all three types of curve can be defined by a focus (or in the case of the ellipse two foci). In the case of a parabola we draw a straight line (the directrix) and choose a point (the focus) and the parabola is the set of points that are an equal distance from the directix and the focus:

(image is from the Wikipedia article linked above).
It should be obvious from the diagram that the focus is at the point where parallel rays hitting a parabolic mirror will converge, so it is both the focus in a mathematical sense and the focal point in an optical sense.
Response to comment:
The centre of curvature is the centre of the osculating circle. We can draw this because near the vertex the parabola looks like part of a circle.
Take the unit circle centred at $(0, 1)$:
$$ (y-1)^2 + x^2 = 1 $$
or expanding this:
$$ y^2 - 2y + 1 + x^2 = 1 $$
If we are very near the origin $y^2 \ll y$ so we can approximate the above expression by:
$$ -2y + 1 + x^2 = 1 $$
which rearranges into the equation for a parabola:
$$ y = \tfrac{1}{2} x^2 $$
And the focus of this parabola is at $(0, \tfrac{1}{2})$. I've used a special case to make the working simple, but you can generalise this to any circle passing centred on the $y$ axis and passing through the origin.
