{
  "title": "Conic section analyzer",
  "human_url": "https://happytails.ai/en/calculators/conic-section-analyzer/",
  "agent_url": "https://happytails.ai/agents/en/calculators/conic-section-analyzer/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/conic-section-analyzer.md",
  "content": "Conic section analyzer\nClassify a second-degree equation in x and y as a circle, ellipse, parabola, hyperbola or degenerate case. Exact invariants, the center, axes, foci and vertices are reported.\nHow can we improve this tool?\nReport a problem or share an idea.\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\nHow to use Conic section analyzer\nType the equation, such as 9x^2 + 4y^2 - 36x + 8y + 4 = 0, or six coefficients A, B, C, D, E, F.\nRun the analysis.\nRead the classification, invariants, center, axes and key points.\nDownload the JSON report or the CSV of points.\nUseful next steps\nGraphing calculator\nQuadratic formula calculator\nVector geometry workspace\nHow the curve is classified\nThe equation is rewritten as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The discriminant B^2 - 4AC and the determinants of the 2 by 2 and 3 by 3 coefficient matrices decide the type, including degenerate cases such as a point, two lines or no real points. Method: OpenStax, Precalculus 2e, section 10.4 Rotation of Axes (2021).\nExact and decimal values\nInvariants, the center and the eigenvalues are exact fractions or square roots. When B is 0 the axes, foci, vertices, asymptotes and directrix are exact too, and the standard form puts a fractional denominator in parentheses, as in y^2/(1/4) = 1, so it reads correctly from left to right. A parabola's vertex, focus and focal length (the distance from vertex to focus, always positive) are exact even when the curve is rotated, as long as each exact value, and every step that produces it, fits in 1,024 digits. Otherwise, they are computed exactly and then rounded to 10 significant digits, marked as not exact. For a rotated ellipse or hyperbola, foci and vertices are decimals with 10 significant digits and are marked as not exact.",
  "content_format": "text/plain",
  "visibility": "public",
  "links": [
    {
      "name": "Graphing calculator",
      "url": "https://happytails.ai/en/calculators/graphing-calculator/"
    },
    {
      "name": "Quadratic formula calculator",
      "url": "https://happytails.ai/en/calculators/quadratic-formula/"
    },
    {
      "name": "Vector geometry workspace",
      "url": "https://happytails.ai/en/calculators/vector-geometry-workspace/"
    }
  ],
  "tool": {
    "id": "conic-section-analyzer",
    "category": "calculators",
    "built": true,
    "requirements": "Equations are limited to 1,000 characters and 100 digits per number, with exact values up to 1,024 digits. Only x, y and numbers are allowed. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of key points.",
    "implementation": {
      "fields": [],
      "registrationFamily": "exact-math-reference",
      "inputMaxLength": 1000,
      "inputErrorCode": "EXACT_MATH_REFERENCE_EXPRESSION_TOO_LONG",
      "note": "Classifies the curve on your device with exact fractions and square roots. Equations are limited to 1,000 characters; rotated ellipses and hyperbolas, and rotated parabolas whose exact points pass 1,024 digits, report foci and vertices as decimals.",
      "experience": {
        "automatic": false,
        "single": true,
        "inputMaxLength": 1000,
        "inputMode": "text",
        "label": "Second-degree equation",
        "placeholder": "9x^2 + 4y^2 - 36x + 8y + 4 = 0",
        "help": "Use x and y with powers up to 2, or six numbers A, B, C, D, E, F. Up to 1,000 characters. Review the settings before running.",
        "action": null
      },
      "mode": "explicit",
      "interactive": false
    },
    "execution": "Browser interface",
    "api": null,
    "browser_agent": {
      "supported": true,
      "name": "run_current_tool",
      "discovery": "WebMCP on the human page in a supporting browser",
      "verification": "See audit/API-AUDIT.md; availability is not verification"
    }
  },
  "markdown": "# Conic section analyzer\n\n- Human page: https://happytails.ai/en/calculators/conic-section-analyzer/\n- Markdown: https://happytails.ai/en/calculators/conic-section-analyzer.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/conic-section-analyzer/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `conic-section-analyzer` |\n| Category | calculators |\n| Implementation | Registered implementation. Registration alone is not a production-quality guarantee. |\n| Browser execution | Use the visible action or interactive controls. |\n| Browser processing | Browser; see tool notes for network behavior. |\n| HTTP API | Not callable through HTTP. |\n| Browser-agent interface | run_current_tool |\n\n### Implementation notes\n\nClassifies the curve on your device with exact fractions and square roots. Equations are limited to 1,000 characters; rotated ellipses and hyperbolas, and rotated parabolas whose exact points pass 1,024 digits, report foci and vertices as decimals.\n\n### Inputs and settings\n\nText input: Second-degree equation.\n\nExample or placeholder shown in the interface (not a validated fixture):\n\n```text\n9x^2 + 4y^2 - 36x + 8y + 4 = 0\n```\n\nUse x and y with powers up to 2, or six numbers A, B, C, D, E, F. Up to 1,000 characters. Review the settings before running.\n\nNo additional settings are declared for this tool.\n\n### Browser-agent workflow\n\n1. Open the human page in a browser that supports WebMCP.\n2. Discover the interface exposed by that page; use its actual schema.\n3. Select files on the page first when the tool requires files.\n4. Call `run_current_tool` with valid input and settings.\n5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links.\n\nCalls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution.\n\n### Planning specification\n\nEquations are limited to 1,000 characters and 100 digits per number, with exact values up to 1,024 digits. Only x, y and numbers are allowed. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of key points.\n\nThis describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support.\n\n### Complete machine-readable capability record\n\n```json\n{\n  \"id\": \"conic-section-analyzer\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"requirements\": \"Equations are limited to 1,000 characters and 100 digits per number, with exact values up to 1,024 digits. Only x, y and numbers are allowed. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of key points.\",\n  \"implementation\": {\n    \"fields\": [],\n    \"registrationFamily\": \"exact-math-reference\",\n    \"inputMaxLength\": 1000,\n    \"inputErrorCode\": \"EXACT_MATH_REFERENCE_EXPRESSION_TOO_LONG\",\n    \"note\": \"Classifies the curve on your device with exact fractions and square roots. Equations are limited to 1,000 characters; rotated ellipses and hyperbolas, and rotated parabolas whose exact points pass 1,024 digits, report foci and vertices as decimals.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": true,\n      \"inputMaxLength\": 1000,\n      \"inputMode\": \"text\",\n      \"label\": \"Second-degree equation\",\n      \"placeholder\": \"9x^2 + 4y^2 - 36x + 8y + 4 = 0\",\n      \"help\": \"Use x and y with powers up to 2, or six numbers A, B, C, D, E, F. Up to 1,000 characters. Review the settings before running.\",\n      \"action\": null\n    },\n    \"mode\": \"explicit\",\n    \"interactive\": false\n  },\n  \"execution\": \"Browser interface\",\n  \"api\": null,\n  \"browser_agent\": {\n    \"supported\": true,\n    \"name\": \"run_current_tool\",\n    \"discovery\": \"WebMCP on the human page in a supporting browser\",\n    \"verification\": \"See audit/API-AUDIT.md; availability is not verification\"\n  }\n}\n```\n\n## Page guide\n\nClassify a second-degree equation in x and y as a circle, ellipse, parabola, hyperbola or degenerate case. Exact invariants, the center, axes, foci and vertices are reported.\n\nHow can we improve this tool?\n\nReport a problem or share an idea.\n\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\n\n### How to use Conic section analyzer\n\n1. Type the equation, such as 9x^2 + 4y^2 - 36x + 8y + 4 = 0, or six coefficients A, B, C, D, E, F.\n2. Run the analysis.\n3. Read the classification, invariants, center, axes and key points.\n4. Download the JSON report or the CSV of points.\n\n## Useful next steps\n\n- [Graphing calculator](https://happytails.ai/en/calculators/graphing-calculator/) ([Markdown](https://happytails.ai/en/calculators/graphing-calculator.md))\n\n- [Quadratic formula calculator](https://happytails.ai/en/calculators/quadratic-formula/) ([Markdown](https://happytails.ai/en/calculators/quadratic-formula.md))\n\n- [Vector geometry workspace](https://happytails.ai/en/calculators/vector-geometry-workspace/) ([Markdown](https://happytails.ai/en/calculators/vector-geometry-workspace.md))\n\n## How the curve is classified\n\nThe equation is rewritten as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The discriminant B^2 - 4AC and the determinants of the 2 by 2 and 3 by 3 coefficient matrices decide the type, including degenerate cases such as a point, two lines or no real points. Method: OpenStax, Precalculus 2e, section 10.4 Rotation of Axes (2021).\n\n### Exact and decimal values\n\nInvariants, the center and the eigenvalues are exact fractions or square roots. When B is 0 the axes, foci, vertices, asymptotes and directrix are exact too, and the standard form puts a fractional denominator in parentheses, as in y^2/(1/4) = 1, so it reads correctly from left to right. A parabola's vertex, focus and focal length (the distance from vertex to focus, always positive) are exact even when the curve is rotated, as long as each exact value, and every step that produces it, fits in 1,024 digits. Otherwise, they are computed exactly and then rounded to 10 significant digits, marked as not exact. For a rotated ellipse or hyperbola, foci and vertices are decimals with 10 significant digits and are marked as not exact.\n"
}