Identification of second order curves

Question

How can I identify a given curve and check whether it is a pair of straight lines, parabola, hyperbola, ellipse or a pair of curves (without plotting it)?

For example....how can I identify the type of the following curve:

9x^2+9y^2+18x+11=0

Is there any specific algorithm which will lead me to the type of curve given?

Answer 1

` ax^2 + bxy + cy^2 + ... = 0

d = b*b-4*a*c

d< 0 - ellipse

d >0 - hyperbola or a pair of straight lines

d = 0 - parabola

`

Answer 2

1.ellipse the coefficients of x² and y² are the same sign and different value ex:10x2+5y2=10

2. hyperbola coefficients of x² and y² have different sign ex:10x2-5y2=10

3. parabola only one is squared x+y2=1 & x2+y=1

4. circle coefficients of x² and y² are the same sign and the same value

   ex:x2+y2=1