Undergrad Hyperbola: Definition & Math Understanding

[parshyaa]

• Circle: its a locus of a point which moves such that its distance from a fixed point is constant
• Ellipse: its a locus of a point which moves such that its distance from two fixed point is constant.
• These definition makes me understand How scientist/mathematican have invented these conics , so what is the best definition of hyperbola , definition which can make me understand the mathematican's approach or thinking.

 

[fresh_42]

The difference between the distance to two fixed points: $| \; ||PF_2|| \, - \, ||PF_1|| \; | = 2 a = const.$

 

[parshyaa]

I know this definition ,but the problem ìs that how can you show that the difference between these distance will give you a two symmetric opposite curve, we can easily realize about ellipse but not with hyperbola.

 

[fresh_42]

The symmetry comes from the absolute value. One branch for each sign of $||PF_2|| - ||PF_1||$. For $a=0$ one gets the degenerate hyperbola, a straight, the height of a double pyramide. Maybe I didn't catch your point.

 

[Svein]

Consult Wikipedia: https://en.wikipedia.org/wiki/Hyperbola

 

[parshyaa]

I think I got

The symmetry comes from the absolute value. One branch for each sign of $||PF_2|| - ||PF_1||$. For $a=0$ one gets the degenerate hyperbola, a straight, the height of a double pyramide. Maybe I didn't catch your point.

 

[robphy]

Here is something interesting from a neat site about geometry:
http://www.cut-the-knot.org/Curriculum/Geometry/CircleAndHyperbola.shtml