Calculating Eccenticity of a Pair of Straight Lines

In summary, the eccentricity of a pair of straight lines is a property of conic sections. However, a pair of straight lines alone cannot form a conic section. The degenerate conic section formed by the pair of lines x/a- y/b= 0 and x/a+ y/b= 0 has an eccentricity of infinity. The correct formula for the eccentricity is \frac{\sqrt{ca^2- cb^2}}{ca}= \frac{\sqrt{a^2- b^2}}{\sqrt{c}a}.
  • #1
lizzie
25
0
how do we define the eccenticity of a pair of straight lines
 
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  • #2
Eccentricity is a property of conic sections. Two lines do not a conic section make.
 
  • #4
but i have read that pair of straight lines is a part of a conic section. how do u define a conic section.
 
  • #5
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
yes.
 
  • #7
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
I think you have the wrong formula for the eccentricity.
 
  • #9
then what is the correct answer
 
  • #10
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.
 

1. What is the formula for calculating the eccentricity of a pair of straight lines?

The formula for calculating the eccentricity of a pair of straight lines is e = √(1 - (b^2/a^2)), where a and b are the lengths of the major and minor axis respectively.

2. How do I know if the eccentricity is a positive or negative value?

If the eccentricity value is less than 1, it is a positive value. If the value is greater than 1, it is a negative value. A value of 0 indicates a circle.

3. Can the eccentricity of a pair of straight lines be greater than 1?

Yes, the eccentricity can be greater than 1. This indicates that the shape formed by the pair of lines is an ellipse.

4. What does an eccentricity of 1 indicate?

An eccentricity of 1 indicates a parabola, where the distance between the two lines is equal at all points.

5. How is the eccentricity of a pair of straight lines related to the shape of the conic section they form?

The eccentricity of a pair of straight lines determines the shape of the conic section they form. A value of 0 indicates a circle, a value between 0 and 1 indicates an ellipse, a value of 1 indicates a parabola, and a value greater than 1 indicates a hyperbola.

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