Parabola - Conic Section and Quadratic Form

Conic Section and Quadratic Form

With reference to the diagram, suppose that there is a horizontal cross-section of the cone passing through P, and that its radius is The angle of inclination of the side of the cone, and also of the plane of the parabola, from the vertical is The chord BC is a diameter of the circle, passing through the point M, which is the midpoint of the chord ED. Let us call the lengths of the line segments EM and DM and the length of PM is

Clearly:

(The triangle BPM is isosceles.)

(PM is parallel to AC.)

Using the intersecting chords theorem on the chords BC and DE, we get:

Rearranging:

For any given cone and parabola, and are constants, so this last equation is a simple quadratic one relating and This shows that the definition of a parabola as a conic section is equivalent to its definition as the graph of a quadratic function. Both definitions produce curves of exactly the same shape. The focal length of the parabola (see below) is It should be noted that many different combinations of and produce the same focal length. In this sense, this situation is different from the one involving the focus and directrix. The focal length of the parabola uniquely specifies the distance between the focus and directrix.