I am guessing that conics with oblique axes are not likely to appear on the SAT. That means there will be no xy term. If that is correct, then you can follow these guidelines:
Circle: Coefficients of x² and y² terms are equal and non-zero.
Ellipse: Coefficients of x² and y² have the same sign, but are unequal.
Hyperbola: Coefficients of x² and y² have opposite signs.
Parabola: Either x² or y² is absent (zero coefficient), but not both.
All of these conditions have degenerate cases in which the curve might be one line, two lines, one point, or nothing at all.