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Diagram
Results
How it’s calculated
Teacher Notes
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 calculator does
This page is for estimating outcomes by changing inputs in one controlled workflow. The model keeps your focus on variables, not output shape. Start with stable assumptions, then test sensitivity by changing one key input at a time to observe directional impact.
Input meaning and unit policy
Each input has an expected unit and a typical range. For reliable interpretation, check whether you are using the same unit system, period, and base assumptions across all runs. Unit mismatch is the most common source of unexpected drift in numeric results.
Use-case sequence
A practical sequence is: first run with defaults, then create a baseline log, then run one alternative scenario, and finally compare only the changed output metric. This sequence reduces cognitive load and prevents false pattern recognition in early experiments.
Common mistakes to avoid
Avoid changing too many variables at once, mixing incompatible data sources, and interpreting a one-time output without checking robustness. A single contradictory input can flip conclusions, so keep each experiment minimal and document assumptions as part of your note.
Interpretation guidance
Review both magnitude and direction. Direction tells you whether a strategy moves outcomes in the desired direction, while magnitude helps you judge practicality. If both agree, you can proceed; if not, rebuild the baseline and verify constraints before deciding.