# Directrix when talking about an ellipse?

1. Mar 22, 2010

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.

#### Attached Files:

• ###### Ellipse.jpg
File size:
22.4 KB
Views:
333
2. Mar 23, 2010

### nuketro0p3r

Re: Directrix?

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

#### Attached Files:

• ###### 424px-Ellipse_parameters.svg.png
File size:
8.7 KB
Views:
851
3. Mar 23, 2010

Re: Directrix?

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;

$$\frac{x^2}{9} \ + \frac{y^2}{16} \ = \ 1$$

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. Mar 23, 2010

### nuketro0p3r

Re: Directrix?

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. Mar 23, 2010

### nuketro0p3r

Re: Directrix?

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. Mar 24, 2010

Re: Directrix?

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

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 it all makes sense now...

Last edited by a moderator: May 4, 2017
7. Mar 24, 2010

### nuketro0p3r

Re: Directrix?

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."