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[
{
"slide": 1,
"fragments": [
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"fragment_index": -1,
"text_description": "Meet the Ellipse\nWhere stretched circles plot the path of planets.",
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]
},
{
"slide": 2,
"fragments": [
{
"fragment_index": -1,
"text_description": "Formal Definition",
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},
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"fragment_index": 1,
"text_description": "Ellipse",
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{
"fragment_index": 2,
"text_description": "An ellipse is all points in a plane whose distances to two fixed foci add to the same constant.",
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"fragment_index": 3,
"text_description": "Quick check: If the constant sum is 10 cm and one focus is 6 cm from point P, how far is P from the other focus?",
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"fragment_index": 4,
"text_description": "Answer: 4 cm",
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}
]
},
{
"slide": 3,
"fragments": [
{
"fragment_index": -1,
"text_description": "Constant-Sum Property",
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{
"fragment_index": 1,
"text_description": "Red segments: \\(PF_1 + PF_2\\) stays constant",
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{
"fragment_index": 2,
"text_description": "One Simple Rule Shapes the Ellipse",
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{
"fragment_index": 3,
"text_description": "An ellipse is the set of all points \\(P\\) for which the sum of distances to two fixed points—the foci \\(F_1\\) and \\(F_2\\)—is constant.",
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{
"fragment_index": 4,
"text_description": "Watch point \\(P\\) move. As \\(PF_1\\) shortens and \\(PF_2\\) lengthens, their combined length never changes, so the path naturally traces the curve.",
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{
"fragment_index": 5,
"text_description": "Key Points:\nTwo fixed points are the foci \\(F_1, F_2\\).\nFor any point \\(P\\), \\(PF_1 + PF_2 = 2a\\) (a constant).\nFixing this sum draws the complete ellipse.",
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}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Parts",
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{
"fragment_index": 1,
"text_description": "Ellipse with labelled axes and vertices",
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{
"fragment_index": 2,
"text_description": "Parts of an Ellipse\nTwo perpendicular axes cross at centre \\(O\\), defining the key components of the ellipse.",
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{
"fragment_index": 3,
"text_description": "Key Points:\nMajor axis: longest diameter; runs through widest points.\nMinor axis: shortest diameter; perpendicular to major axis.\nVertices: four points where axes meet the curve.\nCentre \\(O\\): midpoint where axes cross; symmetry point.",
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]
},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Semi-Lengths & Focus Gap",
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"fragment_index": 1,
"text_description": "",
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"fragment_index": 2,
"text_description": "Relating \\(a\\), \\(b\\) and \\(c\\)\nAn ellipse is governed by three parameters.\nSemi-major \\(a\\) and semi-minor \\(b\\) are half the major and minor axes.\nFocal distance \\(c\\) measures the centre-to-focus gap.",
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{
"fragment_index": 3,
"text_description": "Key Points:\n\\(a \\ge b > 0\\)\nRelation: \\(c = \\sqrt{a^{2}-b^{2}}\\)\nFlatter ellipse when \\(b/a\\) is smaller",
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}
]
},
{
"slide": 6,
"fragments": [
{
"fragment_index": -1,
"text_description": "Why \\(a^{2}=b^{2}+c^{2}\\)?\nGoal: derive the relation linking semi-axes \\(a, b\\) and focal distance \\(c\\).",
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},
{
"fragment_index": 1,
"text_description": "1\nDistance formula\nWrite \\(PF_{1}=\\sqrt{(x+c)^{2}+y^{2}}\\) and \\(PF_{2}=\\sqrt{(x-c)^{2}+y^{2}}\\).",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\nEllipse definition\nSet \\(PF_{1}+PF_{2}=2a\\).",
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},
{
"fragment_index": 3,
"text_description": "3\nSquare & simplify\nIsolate a radical, square twice, then collect like terms.",
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},
{
"fragment_index": 4,
"text_description": "4\nRelation revealed\n\\(x\\) and \\(y\\) cancel, leaving \\(a^{2}=b^{2}+c^{2}\\).",
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{
"fragment_index": 5,
"text_description": "Pro Tip:\nImagine a right triangle with legs \\(b\\) and \\(c\\); its hypotenuse is \\(a\\).",
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}
]
},
{
"slide": 7,
"fragments": [
{
"fragment_index": 1,
"text_description": "Standard Equations",
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},
{
"fragment_index": 2,
"text_description": "Horizontal vs Vertical major axis",
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{
"fragment_index": 3,
"text_description": "Which axis gets \\(a^{2}\\)?",
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{
"fragment_index": 4,
"text_description": "Both forms satisfy \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\) with \\(a \\ge b\\).\nIf \\(a^{2}\\) sits under \\(x^{2}\\), the major axis is horizontal; if under \\(y^{2}\\), it is vertical.",
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},
{
"fragment_index": 5,
"text_description": "Key Points:\nHorizontal major axis: \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\)\nVertical major axis: \\( \\frac{x^{2}}{b^{2}} + \\frac{y^{2}}{a^{2}} = 1\\)",
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}
]
},
{
"slide": 8,
"fragments": []
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{
"slide": 9,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nCorrect!\nExcellent! You correctly related \\(a, b\\) and \\(c\\) and doubled \\(c\\) to get the focal distance.\nIncorrect\nRemember: \\(a\\) is the semi-major axis length 4, \\(b\\) is 3, so \\(c=\\sqrt7\\) and distance between foci is \\(2c\\).\nconst correctOption = 1;\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\nFor the ellipse \\( \\dfrac{x^{2}}{9} + \\dfrac{y^{2}}{16} = 1 \\), what is the distance between its two foci?",
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},
{
"fragment_index": 2,
"text_description": "1\n7 units",
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},
{
"fragment_index": 3,
"text_description": "2\n\\(2\\sqrt7\\) units",
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},
{
"fragment_index": 4,
"text_description": "3\n8 units",
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},
{
"fragment_index": 5,
"text_description": "4\n4 units",
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},
{
"fragment_index": 6,
"text_description": "Hint:\nFirst find \\(c\\) using \\(c^{2}=a^{2}-b^{2}\\); the focal distance is \\(2c\\).",
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},
{
"fragment_index": 7,
"text_description": "Submit Answer",
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}
]
},
{
"slide": 10,
"fragments": [
{
"fragment_index": -1,
"text_description": "Ellipse in a Nutshell\nThank You!\nWe hope you found this lesson informative and engaging.",
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},
{
"fragment_index": 1,
"text_description": "Set of points whose distances to two foci add to a constant.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Standard form: \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 \\) with \\( a > b > 0 \\).",
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},
{
"fragment_index": 3,
"text_description": "Major axis \\(2a\\), minor axis \\(2b\\); foci satisfy \\( c^{2} = a^{2} - b^{2} \\).",
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},
{
"fragment_index": 4,
"text_description": "Eccentricity \\( e = \\frac{c}{a} \\); always between 0 and 1.",
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},
{
"fragment_index": 5,
"text_description": "Reflection property: a line from one focus reflects to the other.",
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},
{
"fragment_index": 6,
"text_description": "Area formula: \\( \\text{Area} = \\pi a b \\).",
"image_description": ""
},
{
"fragment_index": 7,
"text_description": "Next Steps\nAttempt exercise 10.2 and sketch ellipses with varying \\(e\\) to solidify concepts.",
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}
]
}
]