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This is relatively easy to do. The shape of the graph is determined entirely by the coeffiecients and the squared terms.

If there is an x2 term and a y2 term (on the same side of the equation) and the two coeffiecients are equal, then the graph is a circle.

If there is an x2 term and a y2 term (on the same side of the equation) and the two coeffiecients are not equal, then the graph is an ellipse.

If there is only an x2 term or a y2 regardless of the coeffiecients, the graph is a parabola.

If there is an x2 and a y2 term on the same side of the equation and one term is negative, the graph is a hyperbola.

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