ON THE TWO GOLDEN PARABOLA'S

Set Constants:

Alpha equals

And Beta equals

It is well known that the Parabola called the Golden Parabola is described by the equation

let's graph this simple equation.

Let x go from -4 to 4

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

and

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?

Lets try it and see.

We know that in standard position that the equation of the parabola, if it opens to the right, must be .

Using the above coordinates and solving for 2*P we find that 2*P=4, which means that P=2 and our equation of the other parabola is and for simplicity we will let a=1.

We now plot both equation on the same graph.

And we find two points of intersection.

So lets find the second point of intersection.

So the primary root of x is

The other roots are imaginary.

y value is -2*β

Our second point is

We now have the two points of intersection.

The upper point

And the lower point

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

and lets add this to the graph.

It is interesting to note that this focal chord runs through the Focus points of both Golden Parabola's.

Now the length of the focal chord between P and Q is:

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.

So the focus to point P divided by point R to the focus is the Golden Ratio

Now lets find the area between points P and Q bounded by the three curves

After several integrations and simplifications we find the area is:

Which is a function of α

And α and β

Which is 1/2 the area of the rectangle formed by the tangents and normals at points P and Q

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.

Maybe there are more interesting properties than are mentioned in this paper.

REFERENCES :

1. Thomas Koshey, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, Inc., 2001.

2. H. E. Huntly, The Divine Proportion, Dover Publications, Inc., 1970.

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