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[
{
"slide": 1,
"fragments": [
{
"fragment_index": -1,
"text_description": "Constant Sum Magic",
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{
"fragment_index": 1,
"text_description": "",
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"text_description": "Why does the ellipse behave this way?\nVisual intuition: an ellipse is the set of points P whose distances to two foci always add to the same value.\nMove P along the curve; rulers show \\(PF_1\\) and \\(PF_2\\). Their sum stays constant—this is the constant-sum rule.\nKey Points:\nF₁ and F₂ are called the foci.\nFor every P on the ellipse, \\(PF_1 + PF_2\\) is constant.\nString-and-pins demo: string length = major axis (e.g., 12 cm when pins are 8 cm apart).",
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}
]
},
{
"slide": 2,
"fragments": [
{
"fragment_index": -1,
"text_description": "Formal Definition",
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},
{
"fragment_index": 1,
"text_description": "Ellipse",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "An ellipse is the set of all points in a plane for which the\nsum\nof the distances from two fixed points, the\nfoci\n, is constant.",
"image_description": ""
}
]
},
{
"slide": 3,
"fragments": [
{
"fragment_index": -1,
"text_description": "Axes & Parts\nTerminology",
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},
{
"fragment_index": 1,
"text_description": "",
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{
"fragment_index": 2,
"text_description": "An ellipse has two perpendicular axes that meet at the centre.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key Points:\nCentre: intersection point of the two axes.\nMajor axis: longest chord through the centre.\nMinor axis: shortest chord through the centre.\nVertices: endpoints of each axis.\nFoci: two internal points on the major axis.",
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}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": 1,
"text_description": "Key Length Relation",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "\\[a^{2}=b^{2}+c^{2}\\]\nRight-angle geometry inside the ellipse gives this Pythagorean-like link.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Variable Definitions\na\nSemi-major axis\nb\nSemi-minor axis\nc\nFocus distance from centre",
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},
{
"fragment_index": 4,
"text_description": "Applications\nDeriving standard equation\nReplace \\(c^{2}\\) with \\(a^{2}-b^{2}\\) in the distance rule to obtain the canonical form.\nCalculating eccentricity\nUse \\(e=\\frac{c}{a}\\) once \\(c\\) is found from the relation.",
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]
},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Standard Form",
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"text_description": "",
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"fragment_index": 2,
"text_description": "Ellipse centred at origin\nThe canonical equation is \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 \\).\nCentre \\( (0,0) \\); every point \\( (x,y) \\) obeying it lies on the curve.",
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},
{
"fragment_index": 3,
"text_description": "Key Points:\nCalled the\nstandard form\nof an ellipse.\nMajor axis length \\(2a\\) lies along the x-axis.\nMinor axis length \\(2b\\) lies along the y-axis.",
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}
]
},
{
"slide": 6,
"fragments": [
{
"fragment_index": -1,
"text_description": "Deriving x²/a² + y²/b² = 1",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\n\\[PF_{1}+PF_{2}=2a\\]\nStart with the constant focal-distance sum.",
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},
{
"fragment_index": 2,
"text_description": "2\n\\[\\sqrt{(x+c)^{2}+y^{2}}+\\sqrt{(x-c)^{2}+y^{2}}=2a\\]\nInsert distances from \\((-c,0)\\) and \\((c,0)\\).",
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},
{
"fragment_index": 3,
"text_description": "3\n\\[2a\\sqrt{(x-c)^{2}+y^{2}}=4a^{2}-2xc\\]\nRearrange, isolate one radical, then square once.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "4\n\\[a^{2}y^{2}=b^{2}(a^{2}-x^{2})\\]\nSquare again, simplify, use \\(a^{2}=b^{2}+c^{2}\\).",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "5\n\\[\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\]\nDivide throughout; obtain the standard form of an ellipse.",
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},
{
"fragment_index": 6,
"text_description": "Key Insight:\nSuccessive squaring removes radicals; substituting \\(a^{2}=b^{2}+c^{2}\\) converts focus-based terms into the axis lengths \\(a\\) and \\(b\\).",
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}
]
},
{
"slide": 7,
"fragments": []
},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Label the Ellipse\nDrag each term to its correct location to strengthen your ellipse vocabulary.\nCheck\nResults\nconst draggableItems = document.querySelectorAll('.draggable-item');\n const dropZones = document.querySelectorAll('.drop-zone');\n const checkAnswersBtn = document.getElementById('checkAnswersBtn');\n const feedbackArea = document.getElementById('feedbackArea');\n const feedbackContent = document.getElementById('feedbackContent');\n\n draggableItems.forEach(item => {\n item.addEventListener('dragstart', handleDragStart);\n item.addEventListener('dragend', handleDragEnd);\n });\n\n dropZones.forEach(zone => {\n zone.addEventListener('dragover', handleDragOver);\n zone.addEventListener('drop', handleDrop);\n zone.addEventListener('dragenter', handleDragEnter);\n zone.addEventListener('dragleave', handleDragLeave);\n });\n\n function handleDragStart(e) {\n e.target.classList.add('opacity-50');\n e.dataTransfer.setData('text/plain', e.target.dataset.id);\n }\n\n function handleDragEnd(e) {\n e.target.classList.remove('opacity-50');\n }\n\n function handleDragOver(e) {\n e.preventDefault();\n }\n\n function handleDragEnter(e) {\n e.preventDefault();\n e.target.closest('.drop-zone').classList.add('border-green-500', 'bg-green-50');\n }\n\n function handleDragLeave(e) {\n e.target.closest('.drop-zone').classList.remove('border-green-500', 'bg-green-50');\n }\n\n function handleDrop(e) {\n e.preventDefault();\n const dropZone = e.target.closest('.drop-zone');\n dropZone.classList.remove('border-green-500', 'bg-green-50');\n\n const itemId = e.dataTransfer.getData('text/plain');\n const draggedItem = document.querySelector(`[data-id=\"${itemId}\"]`);\n\n if (draggedItem && dropZone) {\n dropZone.appendChild(draggedItem);\n const placeholder = dropZone.querySelector('.text-center');\n if (placeholder) placeholder.style.display = 'none';\n }\n }\n\n checkAnswersBtn.addEventListener('click', () => {\n feedbackArea.classList.remove('hidden');\n feedbackContent.innerHTML = '<p class=\"text-green-600\">Answers checked! Review your results above.</p>';\n });",
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},
{
"fragment_index": 1,
"text_description": "Draggable Items\nFocus\nMajor Axis\nMinor Axis\nCentre\nVertex",
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},
{
"fragment_index": 2,
"text_description": "Drop Zones\nF1 spot\nLong horizontal line\nShort vertical line\nOrigin\nEndpoint of major axis",
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},
{
"fragment_index": 3,
"text_description": "Tip:\nNeed a hint? Recall which part is the longest.",
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}
]
},
{
"slide": 9,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nSubmit Answer\nCorrect!\nExactly! \\(e \\to 0\\) gives a circle.\nIncorrect\nNot quite. \\(e\\) becomes 0 only when both foci coincide — a circle.\nconst correctOption = 0;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('submitBtn');\n const feedbackCorrect = document.getElementById('feedbackCorrect');\n const feedbackIncorrect = document.getElementById('feedbackIncorrect');\n \n let selectedOption = null;\n \n answerCards.forEach((card, index) => {\n card.addEventListener('click', () => {\n answerCards.forEach(c => c.classList.remove('border-blue-500', 'bg-blue-50'));\n card.classList.add('border-blue-500', 'bg-blue-50');\n selectedOption = index;\n });\n });\n \n submitBtn.addEventListener('click', () => {\n if (selectedOption === null) return;\n \n if (selectedOption === correctOption) {\n feedbackCorrect.classList.remove('hidden');\n feedbackIncorrect.classList.add('hidden');\n } else {\n feedbackIncorrect.classList.remove('hidden');\n feedbackCorrect.classList.add('hidden');\n }\n });",
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},
{
"fragment_index": 1,
"text_description": "Question\nWhen the eccentricity \\(e = \\frac{c}{a}\\) tends to 0, the ellipse approaches which shape?",
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},
{
"fragment_index": 2,
"text_description": "1\nA perfect circle",
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},
{
"fragment_index": 3,
"text_description": "2\nA hyperbola",
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},
{
"fragment_index": 4,
"text_description": "3\nA line segment of zero length",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "4\nIts minor axis equals c but shape unchanged",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Hint:\nRemember: both foci merge at the centre when \\(c = 0\\).",
"image_description": ""
}
]
},
{
"slide": 10,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Takeaways",
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},
{
"fragment_index": 1,
"text_description": "Definition: Sum of distances to two foci stays constant.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Parts: major & minor axes, vertices, centre, foci.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key relation: \\(a^{2}=b^{2}+c^{2}\\) links axes to focal length.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Standard form: \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) (horizontal) and its vertical twin.",
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},
{
"fragment_index": 5,
"text_description": "Eccentricity \\(e=\\frac{c}{a}\\) measures oval-ness, \\(0\\le e<1\\).",
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},
{
"fragment_index": 6,
"text_description": "Next Steps\nTry plotting real-world orbits and measuring their eccentricities!",
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},
{
"fragment_index": 7,
"text_description": "Thank You!\nWe hope you found this lesson informative and engaging.",
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}
]
}
]