Latus Rectum

Date: 08/08/2002 at 16:17:39
From: Michelle
Subject: Latus rectum

Dr. Math,

I'm working on quadratic equations, and I'm trying to find the 
definition, an explanation, a formula, or anything that will help me 
to better understand what exactly latus rectum is. So far all I have 
been able to find is that when using the formula (x-h)=4p(y-k), the 
latus rectum is equal to 4p. I don't understand, and any explanation 
would be wonderful!  Thank you.

Michelle

Date: 08/08/2002 at 23:36:18
From: Doctor Peterson
Subject: Re: Latus rectum

Hi, Michelle.

I would think your book would define the term; but then, I've just 
been searching the Web for a picture of a parabola that shows the 
latus rectum, and I can't seem to find any! It's so simple and visual, 
there's no reason not to show it; I'll let you draw it yourself.

Just find a picture of a parabola that shows the focus and either the 
axis or the directrix, or both. Now draw a line through the focus 
that is parallel to the directrix (that is, perpendicular to the 
axis). It will look something like this:


     o            |F           o
    --o-----------o-----------o--
     A o          |         o  B
           o      |      o
                  oV
                  |
                  |
    --------------o-------------- d
                  D

The vertex is V(h,k), the focus is F, the axis is the line DF, the 
directrix is the line d, and the line segment AB is the latus rectum, 
which is Latin for "straight side." Its length is 4p in your equation

    4p(y-k) = (x-h)^2

where p is the focal length DV=VF. That is, the semi-latus rectum FB 
is the same length as FD. If you think about the definition of the 
parabola, you will see why that has to be true.

Here is one of many pages that show a parabola, including the focus 
and directrix, from Eric Weisstein's MathWorld:

      http://mathworld.wolfram.com/Parabola.html 

Here 'a' is used where you use 'p', so don't let that confuse you.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/