Proof.  Let  y  =  mxhe  the  equation  of  any  line  through
the  centre.  Eliminating  y  between  this  equation  and  that  of
the  ellipse,  we  have

a''^~¥~~

from  which  to  determine  the  abscissae  of  the  points  in  which
the  line  intersects  the  ellipse.  Since  this  equation  contains
terms  in  x^  but  none  in  x,  it  will  reduce  to  a  pure  quadratic,
of  which  the  two  roots  are  equal  but  with  opposite  signs.
From  these  roots  we  shall  get,  by  substituting  in  the  equation
y  =  mx,  two  equal  values  of  y  with  opposite  signs.  Hence
the  points  of  intersection  are  at  equal  distances  on  each  side
of  the  origin.
123.  Theorem  III.  The  locus  of  the  centres  of  parallel
chords  of  an  ellipse  is  a  diameter.
Proof.  Let  y  =  mx -{- h  (a)
be  the  equation  of  a  chord;  the
slope  m  being  the  same  for  all
the  chords,  while  h  varies  from
one  chord  to  another.
We  first  find  the  points  of  in-
tersection of  the  chord  with  the  ellipse  in  the  usual  way.
