The curve of the second order is a locus, ax satisfying to the equation²+fy²+2bxy+2cx+2gy+k=0 in which x, y variables, a, b, c, f, g, k - coefficients, and a²+b²+c² is other than zero.
Instruction
1. Lead the curve equation to a kanonichny look. Consider a canonical form of the equation for various curves of the second order: y parabola²=2px; hyperbole x²/q²-y²/h²=1; ellipse x²/q²+y²/h²=1; two crossed straight lines x²/q²-y²/h²=0; point x²/q²+y²/h²=0; two parallel straight lines x²/q²=1, one straight line x²=0; imaginary ellipse x²/q²+y²/h²=-1.
2. Calculate invariants: Δ, D, S, B. For a curve of the second order Δ defines whether the curve is true - nondegenerate or a limit case of one of true - degenerate. Determines by D symmetry of a curve.
3. Define whether the curve is degenerate. Calculate Δ. Δ=afk-agg-bbk+bgc+cbg-cfc. If Δ=0, a curve degenerate if Δ it is not equal to zero, nondegenerate.
4. Find out the nature of symmetry of a curve. Calculate D. D=a*f-b². If it is not equal to zero, the curve has the center of symmetry if it is equal, respectively, has no.
5. Calculate S and B. S=a+f. The invariant In is equal to the sum of two square matrixes: the first with columns a, c and c, k, the second with columns f, g and g, k.
6. Define curve type. Consider degenerate curves when Δ=0. If D> 0, then it is a point. If D
7. Consider nondegenerate curves - it is an ellipse, a hyperbole and a parabola. If D=0, then it is a parabola, its equation of y²=2px where p> 0. If D0. If D> 0, and S0, h> 0. If D> 0, and S> 0, then it is an imaginary ellipse - there is no point on the plane.
8. Choose type of a curve of the second order which suits you. Lead the initial equation if it is required, to a canonical form.
9. Consider for an example y equation²-6x=0. Receive coefficients, proceeding from ax equation²+fy²+2bxy+2cx+2gy+k=0. Coefficients of f=1, c=3, and other coefficients of a, b, g, k are equal to zero.
10. Calculate sizes Δ and D. Receive Δ=-3*1*3=-9, and D=0. It means that a curve nondegenerate as Δ it is not equal to zero. As D=0, the curve has no center of symmetry. On sets of signs, the equation is a parabola. y²=6x.