-x^2 + y^2 -2x -12y + 31 =0
(y^2-12y +36) - (x^2 -2x +1) = -31 +36 -1
(y-6)^2/4 -(x-1)^2/4 =1
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$\displaystyle -x^2 + y^2 - 2x - 12y + 31 = 0$
$\displaystyle y^2 - 12y - (x^2 + 2x) + 31 = 0$
$\displaystyle y^2 - 12y + \left(-6\right)^2 - [x^2 + 2x + 1^2] + 31 - \left(-6\right)^2 + 1^2 + 31 = 0$
$\displaystyle (y - 6)^2 - (x + 1)^2 - 36 + 1 + 31 = 0$
$\displaystyle (y - 6)^2 - (x + 1)^2 - 4 = 0$
$\displaystyle (y - 6)^2 - (x + 1)^2 = 4$
$\displaystyle \frac{(y - 6)^2}{4} - \frac{(x + 1)^2}{4} = 1$.