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Results
Run a calculation to see results.
Standard Form:
—
\\(\\;\\)
| Parameter | Value | Notes |
|---|
Formulas Used
- Standard form (horizontal major): \\(\\dfrac{(x-h)^2}{a^2}+\\dfrac{(y-k)^2}{b^2}=1\\)
- Standard form (vertical major): \\(\\dfrac{(x-h)^2}{b^2}+\\dfrac{(y-k)^2}{a^2}=1\\)
- Relation: \\(c^2=a^2-b^2,\\ e=\\dfrac{c}{a},\\ 0\\le e<1\\)
- Foci (horizontal): \\((h\\pm c,\\ k)\\); Foci (vertical): \\((h,\\ k\\pm c)\\)
- Vertices (horizontal): \\((h\\pm a,\\ k)\\); Co-vertices: \\((h,\\ k\\pm b)\\)
- Directrices (horizontal): \\(x=h\\pm \\dfrac{a}{e}\\); (vertical): \\(y=k\\pm \\dfrac{a}{e}\\)
- Area: \\(A=\\pi ab\\)
- Perimeter (Ramanujan II): \\(P\\approx \\pi (a+b)\\left[1+\\dfrac{3\\eta}{10+\\sqrt{4-3\\eta}}\\right]\\), where \\(\\eta=\\left(\\dfrac{a-b}{a+b}\\right)^2\\)
- Latus rectum length: \\(\\ell=\\dfrac{2b^2}{a}\\)
How to Use
- Choose orientation: major axis horizontal or vertical.
- Enter center \\((h,k)\\).
- Provide any two of \\(a,b,c,e\\). The rest are inferred.
- Press Calculate to generate equations, parameters, and features.
- Copy the equation, or export results to CSV or PDF.
Example Data
| h | k | a | b | Orientation | e | Equation |
|---|---|---|---|---|---|---|
| 0 | 0 | 5 | 3 | Horizontal | 0.8 | (x^2/25) + (y^2/9) = 1 |
| 2 | -1 | 6 | 4 | Horizontal | 0.745356 | ((x-2)^2/36) + ((y+1)^2/16) = 1 |
| -3 | 1 | 7 | 2 | Vertical | 0.958314 | ((x+3)^2/4) + ((y-1)^2/49) = 1 |
| 1.5 | 1 | 4 | 3.5 | Vertical | 0.515388 | ((x-1.5)^2/12.25) + ((y-1)^2/16) = 1 |
Key Relationships Table
Use these identities to derive missing parameters quickly from known pairs.
| Given | Compute | Formula |
|---|---|---|
| a, b | c | \\(c=\\sqrt{a^2-b^2}\\) |
| a, b | e | \\(e=\\dfrac{\\sqrt{a^2-b^2}}{a}\\) |
| a, e | b | \\(b=a\\sqrt{1-e^2}\\) |
| a, e | c | \\(c=ae\\) |
| b, e | a | \\(a=\\dfrac{b}{\\sqrt{1-e^2}}\\) |
| b, c | a | \\(a=\\sqrt{b^2+c^2}\\) |
Parametric Sample Points
For \\(h=0,k=0,a=5,b=3\\) (horizontal major). \\(x=h+a\\cos t,\\ y=k+b\\sin t\\).
| t (degrees) | cos t | sin t | x | y |
|---|---|---|---|---|
| 0° | 1 | 0 | 5 | 0 |
| 30° | 0.866025 | 0.5 | 4.33013 | 1.5 |
| 60° | 0.5 | 0.866025 | 2.5 | 2.59808 |
| 90° | 0 | 1 | 0 | 3 |
| 120° | -0.5 | 0.866025 | -2.5 | 2.59808 |
Eccentricity and Shape Insights
Relationship between eccentricity and aspect ratio \\(b/a\\) for guidance.
| Eccentricity e | b/a | Shape Description |
|---|---|---|
| 0.00 | 1.000 | Perfect circle |
| 0.30 | 0.954 | Very slight ellipse |
| 0.60 | 0.800 | Moderate elongation |
| 0.80 | 0.600 | Pronounced elongation |
| 0.95 | 0.312 | Highly elongated |
FAQs
It is an equation normalized to 1 with squared terms divided by squared semi-axes, centered at \\((h,k)\\), aligned with axes, describing all points \\((x,y)\\) satisfying the relation.
Let \\(a\\) be the semi-major (larger) axis and \\(b\\) the semi-minor. The orientation determines whether \\(a\\) aligns with x or y in the standard equation.
\\(c\\) locates the foci from the center; \\(e=c/a\\) measures “stretch.” Circles have \\(e=0\\); more elongated ellipses have \\(e\\) closer to 1.
Exact perimeter requires an elliptic integral. We use Ramanujan’s second approximation, which is exceptionally accurate for practical engineering and education tasks.
Yes. If you know \\(c\\) and either \\(a\\) or \\(e\\), the calculator derives the remaining parameters and builds the complete standard form.
Absolutely. Choose the vertical orientation, and the equation flips roles of \\(a\\) and \\(b\\) appropriately while adjusting vertices, foci, and directrices.