B Parabola and Hyperbola Question

pbuk

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

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.

kith

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:

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.

DaveC426913

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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Homework Helper
Gold Member
2018 Award
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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"Parabola and Hyperbola Question"

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