Identify the conic section $y = - 2 x y - x + y + 1$

Identify and find the properties of the conic section $y = - 2 x y - x + y + 1$.

The general equation of a conic section is $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$.

In our case, $A = 0$, $B = 2$, $C = 0$, $D = 1$, $E = 0$, $F = -1$.

The discriminant of the conic section is $\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = 4$.

Next, $B^{2} - 4 A C = 4$.

Since $B^{2} - 4 A C \gt 0$, the equation represents a hyperbola.

To find its properties, use the hyperbola calculator.

$y = - 2 x y - x + y + 1$A represents a hyperbola.

General form: $2 x y + x - 1 = 0$A.

Graph: see the graphing calculator.