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[
{
"slide": 1,
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"text_description": "Discover the Ellipse\nFrom planetary paths to classroom graphs—ellipses unveiled.",
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"slide": 2,
"fragments": [
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"fragment_index": -1,
"text_description": "String-Pin Definition",
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"text_description": "Constant-Sum Property\nAn ellipse is the locus of all points \\(P\\) such that \\(PF_{1}+PF_{2}\\) remains constant.",
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"text_description": "Key Points:\nConstant string length gives \\(PF_{1}+PF_{2}=2a\\).\nTwo foci are needed; one focus would form a circle.",
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]
},
{
"slide": 3,
"fragments": [
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"fragment_index": -1,
"text_description": "Naming the Parts",
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"fragment_index": 1,
"text_description": "Standard ellipse with axes, centre, vertices, and foci.",
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"text_description": "Parts of an Ellipse\nAn ellipse has two perpendicular symmetry lines: the major axis and the minor axis.\nThe major axis is the longest chord; its endpoints are called vertices.\nThe minor axis is the shortest chord and meets the major axis at the centre.\nTwo fixed points on the major axis, equidistant from the centre, are the foci.",
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"text_description": "Key Points:\nMajor axis: longest chord, length \\(2a\\).\nMinor axis: shortest chord, length \\(2b\\).\nVertices: endpoints of major axis \\((\\pm a,0)\\).\nCentre: midpoint of both axes \\((0,0)\\).\nFoci: points \\((\\pm c,0)\\) where \\(c^{2}=a^{2}-b^{2}\\).",
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},
{
"slide": 4,
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"fragment_index": -1,
"text_description": "Parameters a, b, c",
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"text_description": "",
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"fragment_index": 2,
"text_description": "Semi-axes and focal distance\nThe ellipse is governed by three parameters: semi-major axis \\(a\\), semi-minor axis \\(b\\) and focal distance \\(c\\).\nThese values define its size, flatness and the position of its two foci.",
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"fragment_index": 3,
"text_description": "Key Points:\nRelationship: \\(b^{2}=a^{2}-c^{2}\\).\nAs \\(c\\) approaches \\(a\\), \\(b\\) decreases and the ellipse stretches.\nIf \\(c=0\\), then \\(b=a\\); the curve becomes a circle.",
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]
},
{
"slide": 5,
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{
"slide": 6,
"fragments": [
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"fragment_index": -1,
"text_description": "Standard Forms",
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{
"fragment_index": 1,
"text_description": "Ellipse with horizontal and vertical axes",
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{
"fragment_index": 2,
"text_description": "Ellipse: choose the right standard equation\nIdentify the major axis, then write the matching standard equation.\nKey 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 \\)\nFor both, \\( b^{2} = a^{2} - c^{2} \\)",
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]
},
{
"slide": 7,
"fragments": [
{
"fragment_index": -1,
"text_description": "e vs b/a Curve",
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{
"fragment_index": 1,
"text_description": "Curve of \\( \\frac{b}{a} = \\sqrt{1-e^{2}} \\)",
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{
"fragment_index": 2,
"text_description": "Eccentricity vs Semi-minor Ratio\nGraph displays the relationship \\( \\frac{b}{a} = \\sqrt{1-e^{2}} \\) between eccentricity and semi-minor ratio.\nAs \\(e\\) rises, \\(b/a\\) falls, flattening the ellipse.",
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{
"fragment_index": 3,
"text_description": "Key Points:\n\\(e = 0\\): \\(b/a = 1\\) — circle\n\\(e = 0.6\\): \\(b/a \\approx 0.8\\) — moderate flattening\n\\(e \\to 1\\): \\(b/a \\to 0\\) — almost a line",
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]
},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "b² = a²(1 – e²)",
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{
"fragment_index": 1,
"text_description": "1\n\\[c = ae\\]\nDefinition of eccentricity: focus distance is e times semi-major axis.",
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{
"fragment_index": 2,
"text_description": "2\n\\[b^{2} = a^{2} - c^{2}\\]\nEllipse axes satisfy a right-triangle relation at any vertex.",
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{
"fragment_index": 3,
"text_description": "3\n\\[b^{2} = a^{2} - a^{2}e^{2} = a^{2}(1 - e^{2})\\]\nSubstitute \\(c = ae\\) and simplify to link b, a, and e.",
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{
"fragment_index": 4,
"text_description": "Key Insight:\nGreater eccentricity shrinks \\(1 - e^{2}\\), so the semi-minor axis shortens while a stays fixed.",
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]
},
{
"slide": 9,
"fragments": [
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"text_description": "Multiple Choice Question",
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"fragment_index": 2,
"text_description": "Question\nFor an ellipse with \\(a = 5\\) and \\(c = 3\\), find \\(b\\).",
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{
"fragment_index": 3,
"text_description": "1\n4\n2\n\\( \\sqrt{16} \\)\n3\n3\n4\n\\( \\sqrt{34} \\)",
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},
{
"fragment_index": 4,
"text_description": "Hint:\nUse \\(b = \\sqrt{a^{2} - c^{2}}\\).",
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{
"fragment_index": 5,
"text_description": "Submit Answer",
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{
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"text_description": "Correct!\nGreat! You applied \\(b = \\sqrt{a^{2}-c^{2}}\\) correctly.\nIncorrect\nRemember \\(b^{2} = a^{2}-c^{2}\\) and try again.\nconst correctOption = 0;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('slide-09-k8m5r2-submit');\n const feedbackCorrect = document.getElementById('slide-09-k8m5r2-feedback-correct');\n const feedbackIncorrect = document.getElementById('slide-09-k8m5r2-feedback-incorrect');\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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{
"slide": 10,
"fragments": [
{
"fragment_index": -1,
"text_description": "Ellipse Essentials\nThank You!\nWe hope you found this lesson informative and engaging.",
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{
"fragment_index": 1,
"text_description": "Locus definition: sum of distances to two foci remains constant.",
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},
{
"fragment_index": 2,
"text_description": "Key parts: centre, foci, vertices, major axis, minor axis.",
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{
"fragment_index": 3,
"text_description": "Parameters satisfy \\(b^{2}=a^{2}-c^{2}\\).",
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{
"fragment_index": 4,
"text_description": "Standard forms: horizontal \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\), vertical \\( \\frac{x^{2}}{b^{2}}+\\frac{y^{2}}{a^{2}}=1 \\).",
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{
"fragment_index": 5,
"text_description": "Eccentricity \\(e=\\frac{c}{a}\\) shows stretch; \\(0<e<1\\).",
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}
]
}
]