Calculator
Diagram
Results
How it’s calculated
Teacher Notes
How to use this calculator effectively
This guide helps you use Conic Sections Explorer in a repeatable way: define a baseline, change one variable at a time, and interpret outputs with explicit assumptions before you share or act on results.
How it works
The page applies deterministic logic to your inputs and shows rounded output for readability. Treat it as a comparison workflow: run one baseline case, adjust a single parameter, and measure both absolute and percentage deltas. If a result seems off, verify units, time basis, and sign conventions before drawing conclusions. This approach keeps your analysis reproducible across teammates and sessions.
When to use
Use this page when you need a fast estimate, a classroom check, or a practical what-if comparison. It works best for planning and prioritization steps where you need direction and magnitude quickly before investing in deeper modeling, manual spreadsheets, or formal external review.
Common mistakes to avoid
- Changing multiple parameters at once, which hides the true cause of output movement.
- Mixing units (percent vs decimal, monthly vs yearly, gross vs net) across scenarios.
- Comparing with another tool without aligning defaults, constants, and rounding rules.
- Using rounded display values as exact downstream inputs without re-checking precision.
Interpretation and worked example
Run a baseline scenario and keep that result visible. Next, modify one assumption to reflect your realistic alternative and compare direction plus size of change. If the direction matches your domain expectation and the size is plausible, your setup is usually coherent. If not, check hidden defaults, boundary conditions, and interpretation notes before deciding which scenario to adopt.
See also
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.