Conic sections

  • Thread starter lizzie
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  • #1
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Main Question or Discussion Point

how do we define the eccenticity of a pair of straight lines
 

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  • #2
Vid
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Eccentricity is a property of conic sections. Two lines do not a conic section make.
 
  • #4
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but i have read that pair of straight lines is a part of a conic section. how do u define a conic section.
 
  • #5
symbolipoint
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Which kind of "pair of lines" do you mean? A pair of intersecting lines can be a conic section (but does this then have an eccentricity?)
 
  • #6
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yes.
 
  • #7
HallsofIvy
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x2/a^2- y2/b^2= 0 is a "degenerate" conic section. It is the limiting case of then hyperbola x2/a2c- y2/a2c= 1 or x2/a2- y2/b2= c as c goes to 0 and, since it can be factored as (x/a- y/b)(x/a+ y/b)= 0, its graph is the two lines x/a- y/b= 0 and x/a+ y/b= 0.

The eccentricy of such a hyperbola is [tex]\sqrt{ca^2- cb^2}{ca}= \sqrt{a^2- b^2}{\sqrt{c}a}[/tex]. As c goes to 0 that goes to 0. Strictly speaking the eccentricity of a degenerate hyperbola is "not defined" but roughly speaking it is infinity.
 
  • #8
Hurkyl
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I think you have the wrong formula for the eccentricity.
 
  • #9
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then what is the correct answer
 
  • #10
HallsofIvy
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Just bad "LaTex". I had meant
[tex]\frac{\sqrt{ca^2- cb^2}}{ca}= \frac{\sqrt{a^2- b^2}}{\sqrt{c}a}[/tex]
and the result is the same as before.
 

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