According to the hint, we first take care of $bxy$ . If $b\neq0$ , then if $a=c=0$ , we can take new coordinates $x+y$ and $x-y$ , if $a\neq0$ , we take $x+\tfrac{b}{2a}y$ and $y$ , if $a=0$ and $c\neq0$ , we take $x$ and $y+\tfrac{b}{c}x$ . Then, we have $a'x^{2}+b'y^{2}$ in new coordinates, and by scaling transform it to $\pm x^{2}\pm y^{2}$ . If $b=0$ , we use scaling before right away. Also, if in the end we have $-x^{2}$ as the very first term, then we can multiply the equation by $-1$ in what follows.

After this, we have $$\begin{eqnarray*} \epsilon_{1}x^{2}+\epsilon_{2}y^{2}+d'x+e'y+f & = & 0, \end{eqnarray*}$$ where $\epsilon_{1}\in\{0,1\}$ and $\epsilon_{2}\in\{-1,0,1\}$ .

If $\epsilon_{1}\neq0$ , we take $x+\tfrac{d'}{2\epsilon_{1}}$ and $y$ to take care of $d'x$ , and similarly if $\epsilon_{2}\neq0$ , we take $x$ and $y+\tfrac{e'}{2\epsilon_{2}}$ to take care of $e'y$ . Therefore, we can further assume that $\epsilon_{1}d'=0$ and $\epsilon_{2}e'=0$ .

If $\epsilon_{1}=0$ and $\epsilon_{2}\neq0$ , we can take $y$ and $x$ as new coordinates, and multiply by $-1$ if necessary, to ensure that if $\epsilon_{1}=0$ then $\epsilon_{2}=0$ .

If $\epsilon_{2}<0$ and $f>0$ , then we take new coordinates $y$ and $x$ to ensure $f<0$ in this case.

If $\epsilon_{2}<0$ and $f=0$ , then we take new coordinates $x-y$ and $x+y$ to transform the equation to $xy=0$ .

If $e'\neq0$ , we can take $x$ and $e'y+f$ to ensure that in this case $e'=1$ and $f=0$ . Similarly, if $d'\neq0$ .

If $\epsilon_{1}>0$ , $\epsilon_{2}=0$ , $e'=0$ and $f<0$ , then we take $x/(2\sqrt{-f})+1/2$ and $y$ to have $x(x-1)=0$ .

If $d'\neq0$ and $\epsilon_{2}=0$ , then we first ensure that $f=0$ as above, and then take $d'x+e'y$ and $y$ to have $x=0$ . Similarly, if $\epsilon_{1}=0=d'$ and $e'\neq0$ , then we first ensure that $f=0$ as above, and then take $e'y$ and $x$ to have $x=0$ .

In summary, we have the cases as in the table below.

$\epsilon_{1}$	$\epsilon_{2}$	$d'$	$e'$	$f$	case
$1$	$1$	$0$	$0$	$>0$	(e)
$1$	$1$	$0$	$0$	$=0$	(d)
$1$	$1$	$0$	$0$	$<0$	(a)
$1$	$0$	$0$	$\neq0$	$=0$	(b)
$1$	$0$	$0$	$0$	$>0$	(f)
$1$	$0$	$0$	$0$	$=0$	(k)
$1$	$0$	$0$	$0$	$<0$	(j)
$1$	$-1$	$0$	$0$	$=0$	(i)
$1$	$-1$	$0$	$0$	$<0$	(c)
$0$	$0$	$\neq0$	$\mathbb{R}$	$=0$	(h)
$0$	$0$	$0$	$\neq0$	$=0$	(h)
$0$	$0$	$0$	$0$	$\neq0$	(g)
$0$	$0$	$0$	$0$	$0$	(l)