It is well known that the Parabola called the Golden Parabola is described by the equation
We find that the Focus is at point F(1/2,-1). The Vertex is at point V(1/2,-5/4). The Directex is at point D(1/2, -3/2). And the distance from the Focus to the Directex P=1/2 .
Now the equation of the Directex is y=-3/2, the equation of the axis is x=1/2, while the length of the Latus Rectum (Focal Width) is 2*P=1 and the roots of this equation are
We now ask. What is the equation of the other Golden Parabola that can be linked to this Golden Parabola in terms of the Golden Section and in terms of connecting alpha and beta with both parabola's.
We don't know, So we proceed by trial and error to find points on this Parabola that might include alpha and beta. We assume the Parabola will be in standard position, that is the Vertex is at (0,0). So the roots are (0,0). After much searching we find that around the point (2.5, 3.2) in the first quadrant we find a point of . Could this be one of the points on the other Golden Parabola that we are looking for?
We find that the Focus of our second Golden Parabola is at point F(1,0). The Vertex is at point V(0,0). The Directex is at point D(-1,0). And the distance from the Focus to the Directex P=2 .
Now the equation of the Directex is x=-1, the equation of the axis is y=0, while the length of the Latus Rectum (Focal Width) is 2*P=4.
If we run a focal chord from point P to Point Q, we find the equation of this focal chord to be
It is interesting to note that this focal chord runs through the Focus points of both Golden Parabola's.
Now the focal chord PQ will intersect the y axis at point R(0,-2). The Focus (F(1,0)) of the right
extending parabola (second parabola) will divide the points R to P in the Golden Ratio.
We note that this area between the two parabola's, if viewed properly, looks like a Fish, aircraft wing, cross section of a whale, cross section of a submarine, a water drop, and maybe other things.
Some very interesting properties are associated with the intersection of both parabola's and the focal chord P to Q. These are investigated in the books by Thomas Koshy, Fibonacci and Lucas Numbers with Applications, and, H. E. Huntly, The Divine Proportion.