Here's how you tell just by looking at the equation, assuming there are no xy terms:

Circle -- 2nd degree polynomial equation having both x² and y² terms with the
same COEFFICIENT when on the same side of the equation.

Parabola -- 2nd degree polynomial equation having only an x² term or a y² term
but not both.

Hyperbola -- 2nd degree polynomial equation having both x² and y² terms with
OPPOSITE SIGNS when on the same side of the equation.

Ellipse -- 2nd degree polynomial equation having both x² and y² terms with
DIFFERENT COEFFICIENTS BUT BOTH WITH THE SAME SIGN when on the same side of the
equation.

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x²-4y²-4x-24y=48

That's a polynomial equation having both x² and y² terms with OPPOSITE SIGNS
when on the same side of the equation.  So it's a HYPERBOLA.

Edwin