



Parabolas
do have focus points,



but since
a parabola only has one "U," 


it only
has one focus point. 





Example: 


Find the
focus point on the parabola: 





Y
= 2X^{2}












There are
two steps to the process. 


The first
is to move the coefficients to the Y 


The
second step is to factor a 4 out of the Y coefficient 


(who
knows why a 4 works for this, but it does) 


What's
left as the Y coefficient is the distance 





So: 


Step 1 








Step 2: 








So the
distance from the bottom of the "U" of the parabola 


to the
focus point is ^{1}/8 unit. 











Let's try
another one ... 





Example: 


Find the
focus point of the equation: 


Y
= 8(X  3)^{2} + 2












Step 1: 


Move the
8 away from the X ^{2} term 











Step 2: 


Factor
the mysterious 4 out of the Y coefficient 











The
distance from the bottom of the "U" (called the vertex) 


to the
focus point is ^{1}/32. 











So what
does this focus point do? 





By
itself, not much. 


But let's
make it do something of cosmic significance. 


OK, maybe
it's not THAT big a deal ... 





Draw a
line below the vertex of the parabola 


the same
distance as the distance to the focus. 











Here's
the deal. 


From any
point on the parabola, 


it's the
same distance to the focus point 


as it is
to the closest point on the line we just drew ... 











OK, OK,
this seemed like a real yawner to me when I learned it too, 


but
there's a bunch of stuff in Physics that actually uses these things. 


Those
tricks are enough to learn at one time 


so we
show you the math part now. 


Then when
you get to the Physics stuff, it isn't all 


hitting
you at one time. 





Not going
on to physics you say? 


Don't
care about math and science? 





OK, No
problem. 


You can
practice for a career without them 


right
here and now. 





Just
repeat after me: 





"Do
you want ..." 





"fries
with that?" 





Still
with me? 


OK, back
to parabolas ... 





If the X^{2}
term has a minus sign in front of it 


the
"U" is upside down. 


That
would make the focus point below the parabola vertex 


and the
horizontal line above it ... 











But
everything would still work the same. 





We could
even reverse the X and Y. 





Example: 


Find the
focus point of: 


X
= 2Y^{2}












Since
this one is "sideways," 


move the
coefficient TO the X term 


and
factor out the magic 4. 











So the
focus point is ^{1}/8 unit to the right 


of the
vertex of the "U" 


And the
line is ^{1}/8 unit to the left 


of the
vertex of the "U." 











Math
types didn't like calling the line "The Line." 


They like
fancy names that sound important. 


So they
came up with a fancy name for the line. 


They call
it: 


THE
DIRECTRIX






Whatever! 





copyright 2005 Bruce Kirkpatrick 
