B Parabola and Hyperbola Question

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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!
 

DaveC426913

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

Here it is again:
243403


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.
 

kith

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No parabola is simultaneously a hyperbola simply because it doesn't fit the definition. But parabolas can be seen as limiting cases of both ellipses and hyperbolas. Examples:
 
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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.
 

DaveC426913

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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.
 
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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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Charles Link

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The hyperbola of ## xy=1 ## was introduced above. I don't think this was previously mentioned, and it may be worth mentioning, that it will take the standard form of a hyperbola ## \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 ## if you introduce a rotation of axes by ## \theta=45^{\circ} ## using ## \\ ## ## x'=x \cos{\theta}+y \sin{\theta} ## ## \\ ## and ## \\ ## ## y'=-x \sin{\theta}+y \cos{\theta} ##. ## \\ ## You then solve for ## x ## and ##y ## and substitute into ## xy=1 ##. ## \\ ## The result is ## \frac{x'^2}{2}-\frac{y'^2}{2}=1 ##.
 
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