at least one of is not 0.
WLOG, let .
consider the transformation:
case2: at least one of is not 0.
and another WLOG, let .
arrive at: ...
case4: from , at least one of is not = 0
If , by interchanging and , we can force .
now, WLOG, let
by this transformation, you shall arrive at:
you arrive at:
so from there, consider all the combinations of plus and minus there..
you arrive at: or simply
well, the discussion is too long so try to understand it by yourself.. and this is just a guide or outline of the whole proof..
and another: the reason why there are 7 cases is that, this actually shows that any second degree polynomial in the affine plane can be transformed into one of the following:
1. empty set
2. single point
3. a doubled line i.e. two lines which are coinciding
4. two distinct lines
5. a conic..
you shall see which cases gives you the conics (which you are looking for)..