Directrix when talking about an ellipse?

  • #1
sponsoredwalk
532
5
Hey what is the directrix when talking about an ellipse?

I found a book that finally shed light on what eccentricity was, but there's still no mention of what the directrix is.

To be honest, it doesn't even explain eccentricity but I intuitively understand it, however the directrix is something that I don't understand with reference to an ellipse, nor how to find it nor why anybody wants to.

Perhaps you could shed some light on how to go about finding it.
 

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Answers and Replies

  • #2
nuketro0p3r
12
0


If I am not mistaken, the directrix is not drawn in that figure. This might help.
 

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  • #3
sponsoredwalk
532
5


Thanks, I know what the directrix is it's just that I don't know how to calculate it given an equation (nor why you'd even want to). I don't see the relationship to anything discussed when talking about ellipses (maybe why It took me so long to even see it in a book).

Take an equation;

[tex] \frac{x^2}{9} \ + \frac{y^2}{16} \ = \ 1 [/tex]

I calculate the focus(es) to be;

c = ±√(a² - b²) = ±√(16 - 9) = ±√7

Where a is the semi-major axis (4²) and b is the semi-minor axis (3²) here.

The eccentricity to be;

e = c/a = (√7)/4

But how do I calculate the directrix, and why would I want to?

That's all I'd really like to know, if anybody knows why please let me know.

thanks :)
 
  • #4
nuketro0p3r
12
0


For an ellipse the equation of directrix is given by x=± a/e. It's been a while since I've studied them and its late here so I have to go to bed now :). I'll revise my course tomorrow and try to find out the answer to that tomorrow.
 
  • #5
nuketro0p3r
12
0


Meanwhile this might help.
"The directrix of a conic section is the line which, together with the point known as the focus, serves to define a conic section as the locus of points whose distance from the focus is proportional to the horizontal distance from the directrix, with r being the constant of proportionality. If the ratio r=1, the conic is a parabola, if r<1, it is an ellipse, and if r>1, it is a hyperbola"
 
  • #6
sponsoredwalk
532
5


Thanks for your responses,

After your reply I understand the following quote form wikipedia a little better;

Each focus F of the ellipse is associated to a line D perpendicular to the major axis (the directrix) such that the distance from any point on the ellipse to F is a constant fraction of its distance from D. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse. The ratio between the two distances is the eccentricity e of the ellipse; so the distance from the center to the directrix is a/e.


So I went ahead and made a "paint" ellipse and this is what came out of it;

http://img10.imageshack.us/img10/8403/directrix.jpg [Broken]

Then I measured the ratio of the line from the focus to the point & the directrix to the point & you get the eccentricity.

It all makes sense now!

Thanks :biggrin: it all makes sense now...
 
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  • #7
nuketro0p3r
12
0


Nice, here's another definition I found while looking for a better explanation.
"A Conic is the set of all the points whose distance from a fixed point bears a constant ratio to its distance from a fixed line. The fixed point is called the focus, the fixed line is called the directrix and the constant ratio is known as the eccentricity of the conic."
 

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