Help classifying conic sequences?

Question

When given a, b and c, how do you find whether the conic sequence is a parabola, circle, eclipse or hyperbole? I think I remember learning something about "b" being greater than zero etc...

Ex: 7x^2 - 12y^2 - 14x + 24y - 28 = 0

Answer 1

x^2/a^2 - y^2/b^2= ...hyperbolic

x^2/a^2 + y^2/b^2= ..... ellipse ... unless a&b are = [circle]

Answer 2

Using just a, b, and c:

∄ b, c  ⇒  parabola

c² = a² + b²  ⇒  hyperbola

c² = a² - b²  ⇒  ellipse  (by convention, a>b)

c² = a² - b²  AND a²=b²  ⇒  circle