Can Parabolas Transform into Ellipses?

[Tris Fray Potter]

I know the difference between the two, but I was wondering if parabolas ever became so steep that they turned back into an elliptical shape.

 

[Drakkith]

Nope. The 'arms' of a parabola continue to get further apart forever.

 

[fresh_42]

I know the difference between the two, but I was wondering if parabolas ever became so steep that they turned back into an elliptical shape.

Well observed, indeed. They are different sections of the same double cone, only at different angles:
https://en.wikipedia.org/wiki/Conic_section

Personally, I find the first image in this version better to see what it is about:
https://de.wikipedia.org/wiki/Kegelschnitt

 

[symbolipoint]

No. Different defined descriptions also. Look at the typical definitions as learned through Intermediate Algebra. Parabola uses ONE focus; ellipse uses TWO foci.

 

[Tris Fray Potter]

Well observed, indeed. They are different sections of the same double cone, only at different angles:
https://en.wikipedia.org/wiki/Conic_section

Personally, I find the first image in this version better to see what it is about:
https://de.wikipedia.org/wiki/Kegelschnitt

Thank-you! I've only worked with parabolas on a Cartesian plane, so I didn't know that it was part of a cone, and I couldn't decipher anything I found when I did some research!

 

[symbolipoint]

Thank-you! I've only worked with parabolas on a Cartesian plane, so I didn't know that it was part of a cone, and I couldn't decipher anything I found when I did some research!

The necessary research is just studying Algebra at the intermediate level, and you will find the most appropriate instruction, textbook discussions, and exercises. Parabola has its own definition using the distance formula. Ellipse has its own but different definition using the distance formula. The definitions and the distance formula are used in deriving equation of each shape. You will want a good instructional textbook on Intermediate Algebra.

 

[Martin Rattigan]

The answer is almost. Look at the focus/directrix definition of conics http://mathworld.wolfram.com/ConicSection.html. Parabolas are defined as those conics with eccentricity 1. Anything below and it's defined to be an ellipse. Anything above and it's defined to be a hyperbola. It's easy to prove that 1 is the only value where the curve doesn't meet the axis again (at least not this side of infinity).

(You might find the $e=0$ case a little confusing.)

 

[sudhirking]

I was wondering if parabolas ever became so steep that they turned back into an elliptical shape.

Regardless of the connections these ellipses have with parabolas, and how they are all conceivably unified under a similar theme, this question must be answerable as a negative. For we know the expression of a parabola, 0 = Ax² + Bx¹ + Cx⁰ - y. It is clearly a function in the sense that any input x gets assigned to a single output y. No matter how the parameters A,B, or C are tuned, this quality, of the parabola being a function, is unchanged.

That ability you described, "to become so steep that they turn back into themselves", that is signature to the other conics, which are curves that are not functions. They have that ability because the forms of the equations are where we see no explicit solution for y, but we see combinations of powers of x and y. For instance, a circle is 0 = x² + y² - A. The y appears squared.

The way all these conics are connected is as follows. Considers functions ƒ(x,y) = 0 where we take these functions to be sums of products of powers of x with powers of y of upto an order 2. By this I mean consider the products xy, y², or x². These are considered order 2 terms. Order 1 terms are x or y. Order 0 terms are constants. Anyways think of taking combinations of these different ordered terms and equating it to 0. The resulting collection of pairs (x,y) that satisfy the equation is called a conic.

 

[Mark44]

Parabolas are one of several kinds of curves called conic sections. The other conic sections are ellipses, circles, and hyperbolas. These curves are called conic sections because they can be obtained by taking different cross sections of a cone. (Note that a mathematical cone differs from the usual notion of a cone in that it is generated by revolving a line around a different line where the two lines aren't parallel or perpendicular. Consequently a mathematical cone has an upper half cone and a lower half cone.)
The various cross sections can be visualized by imagining that a knife is used to cut through the cone. If the cut is perpendicular to the axis of the cone, the curve at the cut is a circle. The cross section made by a vertical cut (parallel to the cone's axis) defines the two sheets of a hyperbola. A cut made parallel to the line that is revolved to make the cone defines a cross section that is a parabola. Finally, a cut made that isn't vertical, isn't perpendicular to the cone's axis, and isn't parallel to the line that generated the cone defines an ellipse cross section.
Here's a sketch of part of a mathematical cone -- it actually extends upward and downward, but I don't show that.

The general equation of any conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Some of the constants A through F can be zero, which gives you the various conic sections.

 

[jbriggs444]

A hyperbola [not an ellipse] is also generated by a slice that is is not vertical but is somewhere between parallel with the edges of the cone and parallel to the axis.