Equations of a Parabola

The Parabola is a geometry figure.   Use one of the geometric definitions of a parabola and derive equations for the parabola.

A.   Geometric definition:    The parabola is a conic figure formed by the intersection of a cone and a plane when the plane is parallel to the generator (side) of the cone.

How does one begin with this 3-dimensional definition of the parabola an arrive at either the 2 dimensional definition or the usual equations of the parabola?




B.    Geometric definition:   In a plane (or a parallel plane) the Parabola is the set of points equidistant from a fixed point called the Focus and a fixed line called the Directrix.

How does one connect the three dimensional definition of the parabola as a conic section to this two dimensinal definition?

How does one derive Cartesian equations of the parabola from this geometric definition?



Introduce an appropriate coordinate system.  Suggestion: Place the (0,0) at the vertex of the parabola, a point midway between the Focus and the Directrix.  The distance   p  from the focus to the vertex is the focal length of the parabola. Place the y axis through the Focus.   Then consider a point (x, y) on the parabola.  We know from the definition that the distance from the focus to (x,y) is the same as the distance from (x,y) to the directrix.   The focus is at (0, p) and the point on the directrix is (x,-p).  Using the distance formulas, we have


So, here we have an equation for the basic parabola where the coefficient of the x-term is a function of the focal length.