Define hyperbola

  • #1
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  • 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.
 

Answers and Replies

  • #2
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The difference between the distance to two fixed points: ##| \; ||PF_2|| \, - \, ||PF_1|| \; | = 2 a = const.##
 
  • #3
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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
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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.
 
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  • #6
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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.
 

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