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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 Copy URL to share inputs, Copy LaTeX for formulas, and Export CSV for sampled points from the plotted curve.
How to use Conic Sections Explorer effectively
What this page is for
Use this page to compare circle, ellipse, parabola, and hyperbola parameters from a clear equation or geometry setup. Start from the form you have, then read center, axes, focus, directrix, and eccentricity outputs in that same context.
Input checks
Confirm whether the equation is standard, general, or vertex form before converting. Small sign changes can move the center or flip the opening direction, so keep coefficients and units visible in your notes.
Workflow
A useful sequence is to identify the conic type, normalize the equation, inspect the key geometric values, and then test one coefficient change. This keeps algebraic changes tied to visible geometry.
Common mistakes
Do not mix equation forms or assume every coefficient set describes a real conic. If the displayed geometry conflicts with expectations, recheck signs, squared terms, and constants before drawing conclusions.
How to read the result
Interpret both the shape and the reference points. Axes, foci, asymptotes, and directrix values explain why the graph behaves the way it does and help catch copied-equation errors.