Work out conic section identifier instantly with clear inputs, formula shown and shareable results.
For the general second-degree curve Ax² + Bxy + Cy² + Dx + Ey + F = 0, the sign of the discriminant B² − 4AC fixes the type: negative gives an ellipse (a circle when A = C and B = 0), zero a parabola, positive a hyperbola.
Discriminant test
B² − 4AC < 0 ellipse, = 0 parabola, > 0 hyperbola
Axis rotation
θ = ½·atan2(B, A − C)
A non-zero xy term means the conic's axes are rotated relative to the coordinate axes; the reported angle removes that rotation.
No. They only translate the curve, though they can make it degenerate into a point, a line or nothing at all.