% x-axis ellipse problem

off rlabout;



rlset reals;


{}


ell := all({x,y},b**2*(x-c)**2+a**2*y**2-a**2*b**2=0 impl x**2+y**2-1 <=0);


                        2  2    2  2    2  2      2        2  2
ell := all x all y ( - a *b  + a *y  + b *c  - 2*b *c*x + b *x  = 0

       2    2
 impl x  + y  - 1 <= 0)


rlgqe ell;


{{a <> 0,

  a - b <> 0,

  a + b <> 0},

             2            2               2            2
 b <> 0 and a  - 2*a*c + c  - 1 <= 0 and a  + 2*a*c + c  - 1 <= 0 and (

  2  2    2    4    2  2    2           2    2
 a *b  - a  - b  - b *c  + b  > 0 or ((a  - b  <= 0

      2  2    2    4    2  2    2
  or a *b  - a  - b  - b *c  + b  <> 0 or ((

  4      2  2      2  3    2      2  3    2
 a *c - a *b *c - a *c  - a *c - b *c  + b *c > 0

  or (a - c - 1 <> 0 and a - c + 1 <> 0 and a + c - 1 <> 0 and a + c + 1 <> 0)

       2      2             2  2    2  2    2    4    2  2    2
  or (a *c - b *c >= 0 and a *b  + a *c  - a  - b  - b *c  + b  >= 0)) and (

              4      2  2      2    4      2
 (c <= 0 and a  - 2*a *c  - 2*a  + c  - 2*c  + 1 <= 0) or (

  4      2  2      2    4      2
 a  - 2*a *c  - 2*a  + c  - 2*c  + 1 >= 0

       4    2  2    2  2    2    2  2    2
  and a  - a *b  - a *c  - a  - b *c  + b  <= 0)))) and (

   2    2           2  2    2    4    2  2    2
 (a  - b  <= 0 and a *b  - a  - b  - b *c  + b  = 0) or ((

  4      2  2      2  3    2      2  3    2
 a *c - a *b *c - a *c  - a *c - b *c  + b *c < 0

  or (a - c - 1 <> 0 and a - c + 1 <> 0 and a + c - 1 <> 0 and a + c + 1 <> 0)

       2    2           2  2    2  2    2    4    2  2    2
  or (a  - b  >= 0 and a *b  + a *c  - a  - b  - b *c  + b  <= 0)

       2      2             2  2    2  2    2    4    2  2    2
  or (a *c - b *c >= 0 and a *b  + a *c  - a  - b  - b *c  + b  >= 0)) and (

              4      2  2      2    4      2
 (c >= 0 and a  - 2*a *c  - 2*a  + c  - 2*c  + 1 <= 0) or (

  4      2  2      2    4      2
 a  - 2*a *c  - 2*a  + c  - 2*c  + 1 >= 0

       4    2  2    2  2    2    2  2    2
  and a  - a *b  - a *c  - a  - b *c  + b  <= 0))))))}


end;

