Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3x2 + y + 2 = 0?

It is a parabola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: 2x2 +3xy - 4y2 + 2x - 3y + 1 = 0?

It is a hyperbola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: 2x2 -3y2 = 0?

It is a hyperbola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3x2 + xy - 2y2 + 4 = 0?

It is an ellipse or a circle.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: x = 0?

It is a degenerate conic. x = 0 is a line.
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