Can the b value in an equation of a hyperbola be 0?

Question

My teacher asked us to comment on possible values of the numerical eccentricity of hyperbolas. I came to a conclusion, and also saw online, that the value range of ɛ is ⟨1, +inf].

ɛ = √(a² + b²) / a

For ɛ < 1, b² would have to b² < 0 which is impossible with real values of b. When b² > 0 ɛ is ɛ > 1. However, when b² = 0 ɛ is ɛ = 1, but is that even a hyperbola then?

Answer 1

The eccentricity characterises a conic section the following way:

• A circle has eccentricity $0$
• An ellipse has eccentricity between $0$ and $1$
• A parabola has eccentricity $1$
• A hyperbola has eccentricity strictly greater than $1$

Geometrically, conic sections are, well, cross sections of a cone. And the eccentricity is the ratio between the sine of the slope of the intersecting plane with the sine of the slope of the cone, assuming the cone has a vertical axis (and horizontal is angle $0^\circ$).

If you're using your formula with semi-major and semi-minor axes, then for an ellipse, the formula for eccentricity is almost the same, except you subtract $a^2-b^2$ in the square root.

Answer 2

If you want to call the cone planar intersection curve a hyperbola, then the least value admissible for eccentricity is 1. This happens when cutting plane is parallel to a cone generator.

$\varepsilon =0 $ is a circle, not even a parabola... much less a hyperbola.