It is impossible to define a parabola with only two points given. An infinite number of parabolic functions will share the two points. Remember that a parabolic function [ax2 + bx1 + cx0] is defined by three coefficients: a, b and c. Two given points only allow you to form two equations with these coefficients. Three equations are generally needed to solve simultaneously for three variables, so three points need to be supplied to pin down your rule/function.
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One definition of a parabola is the set of points that are equidistant from a given line called the directrix and a given point called the focus. So, no. The distances are not different, they are the same. The distance between the directrix and a given point on the parabola will always be the same as the distance between that same point on the parabola and the focus. Any point where those two distances are equal would be on the parabola somewhere and all the points where those two distances are different would not be on the parabola. Note that the distance from a point to the directrix is definied as the perpendicular distance (also known as the minimum distance).
Any three given points can be joined by a common plane, and any two given points can be joined by a common line and an infinite number of common planes.
line
Straight line
In three dimensions, the solid defined as being bound by the set of points at a given distance form a point is a sphere. In two dimensions, the figure defined as being bound by the set of points at a given distance from a point is a circle. In one dimension, a line segment is bound by the two points at a given distance from a point.