# Conic sections

## Main Question or Discussion Point

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

Vid
Eccentricity is a property of conic sections. Two lines do not a conic section make.

Vid

but i have read that pair of straight lines is a part of a conic section. how do u define a conic section.

symbolipoint
Homework Helper
Gold Member
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?)

yes.

HallsofIvy
Homework Helper
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 $$\sqrt{ca^2- cb^2}{ca}= \sqrt{a^2- b^2}{\sqrt{c}a}$$. 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.

Hurkyl
Staff Emeritus
Gold Member
I think you have the wrong formula for the eccentricity.

then what is the correct answer

HallsofIvy
$$\frac{\sqrt{ca^2- cb^2}}{ca}= \frac{\sqrt{a^2- b^2}}{\sqrt{c}a}$$