# Define hyperbola

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

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.

fresh_42
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.

• 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.