how to quickly judge which is an ellips hyperbola or parabola

Circle. When x and y are both squared and the coefficients on them are the same — including the sign.

For example, take a look at 3x2 – 12x + 3y2= 2. Notice that the x2 and y2 have the same coefficient (positive 3). That info is all you need to recognize that you’re working with a circle.

Parabola. When either x or y is squared — not both.

The equations y = x2 – 4 and x = 2y2 – 3y + 10 are both parabolas. In the first equation, you see an x2 but no y2, and in the second equation, you see a y2 but no x2. Nothing else matters — signs and coefficients change the physical appearance of the parabola (which way it opens or how fat it is) but don’t change the fact that it’s a parabola.

Ellipse. When x and y are both squared and the coefficients are positive but different.

The equation 3x2 – 9x + 2y2 + 10y – 6 = 0 is one example of an ellipse. The coefficients of x2 and y2 are different, but both are positive.

Hyperbola. When x and y are both squared, and exactly one of the coefficients is negative and exactly one of the coefficients is positive.

The equation 4y2 – 10y – 3x2 = 12 is an example of a hyperbola. This time, the coefficients of x2 and y2 are different, but exactly one of them is negative and one is positive, which is a requirement for the equation to be the graph of a hyperbola