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FAQ
How do I classify a conic from Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0?
Use the discriminant Δ=B^2−4AC. If Δ<0: ellipse (A=C ⇒ circle), Δ=0: parabola, Δ>0: hyperbola.
How is the rotation angle chosen?
We use θ = ½·atan2(B, A−C) to eliminate the xy term, then round tiny residuals to zero for numerical stability.
How do I obtain the standard form and parameters (a, b, p)?
After rotation, solve [2A' B'; B' 2C']·[X0;Y0] = −[D';E'] to find the center (or vertex). Translate to remove linear terms and normalize to circle/ellipse/hyperbola forms; for a parabola, complete the square to identify p in u^2=4pv.
How are foci, directrices, and asymptotes computed?
Ellipse: c=√(|a^2−b^2|), e=c/max(a,b); directrices are at ±a/e along the major axis. Hyperbola: c=√(a^2+b^2), e=c/a; asymptotes v=±(b/a)u mapped back after rotation/translation. Parabola u^2=4pv has focus (0,p) and directrix v=−p.
Can I share or export my results?
Yes. Use URL কপি to share inputs, Copy LaTeX for formulas, and Export CSV for sampled points from the plotted curve.