Parabola and Hyperbola Question

[Mark44]

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.

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.

 

[Charles Link]

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 $\frac{x'^2}{2}-\frac{y'^2}{2}=1$.