# I Define hyperbola

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1. Jul 3, 2016

### 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.

2. Jul 3, 2016

### Staff: Mentor

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

3. Jul 3, 2016

### 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 realise about ellipse but not with hyperbola.

4. Jul 3, 2016

### Staff: Mentor

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.

5. Jul 3, 2016

6. Jul 4, 2016

### parshyaa

I think I got

7. Jul 7, 2016