Parabola and Hyperbola Question

pbuk
Gold Member
The red cross-section forms a hyperbola, but the blue does not? It forms a parabola instead? They are two different animals?
The red section forms an ellipse!

Gold Member
The red section forms an ellipse!
You were too quick! I saw the error right away and corrected the pic.

Here it is again:

The blue line forms a parabola. The red line forms a hyperbola.
The angle between them could be infinitesimally small.

I would have thought that a hyperbola with E=∞ would be the same animal as a hyperbola with E=∞-1. Just like a circle with E=0 is the same animal as an ellipse with E=0.0000000000000000001.

But OK., I'll let that go.

Mark44
Mentor
I would have thought that a hyperbola with E=∞ would be the same animal as a hyperbola with E=∞-1.
∞-1 is not defined. You can't use infinity in arithmetic computations.

Gold Member
You can't use infinity in arithmetic computations.
Nor am I doing so. I am conceptualizing an arbitrarily large number.

You must grant that a hyperbola can have an eccentricity of an arbitrarily large value - there is no value on the real number line greater than 1 that a hyperbola cannot have as its eccentricity. Literally, anything shy of infinity.

Mark44
Mentor
Nor am I doing so. I am conceptualizing an arbitrarily large number.
Actually, you did use ∞ in an arithmetic expression. You wrote E=∞-1 in post #27, to which I replied that ∞ - 1 is meaningless, as is any arithmetic expression involving the symbol ∞.

Having said that, some expressions using this symbol are meaningful, such as ##\infty + \infty## or ##\infty \times \infty##, which should really be thought of as the sum or product, respectively, of functions whose limits are ##\infty##.

Other arithmetic expressions, such as ##\infty - \infty##, ##\infty - n##, where n is any finite number, and ##\frac \infty \infty## are not defined, and are therefore meaningless.
DaveC426913 said:
You must grant that a hyperbola can have an eccentricity of an arbitrarily large value - there is no value on the real number line greater than 1 that a hyperbola cannot have as its eccentricity. Literally, anything shy of infinity.
But all you need to say is that the eccentricity is greater than 1.

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