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"slide": 1,
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"text_description": "The Ellipse\nWhen circles reach for the horizon.",
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"slide": 2,
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"text_description": "All Points with Equal Sum",
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"text_description": "Ellipse: points whose distances to two foci add to a constant.",
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"text_description": "Geometric Definition of an Ellipse\nPlace two fixed points called foci, \\(F_1\\) and \\(F_2\\).\nAny point \\(P\\) where \\(PF_1 + PF_2\\) stays constant lies on the ellipse.\nThus, an ellipse is the locus defined by this constant-sum rule.\nKey Points:\nConstant value must be greater than the distance \\(F_1F_2\\).\nCloser foci make the ellipse rounder; farther foci stretch it.\nQuiz: Which everyday running track follows this two-focus rule?",
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{
"slide": 3,
"fragments": [
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"fragment_index": -1,
"text_description": "Name the Parts",
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"text_description": "",
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"text_description": "Locate each feature on an ellipse\nFind the centre first, then use it to spot every other part.\nKey Points:\nMajor axis: longest chord through the centre.\nMinor axis: shorter chord perpendicular to the major axis.\nVertices: endpoints of each axis.\nCentre: intersection of the two axes.\nFoci: two fixed interior points on the major axis.",
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{
"slide": 4,
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"fragment_index": -1,
"text_description": "Meet a, b and c",
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"text_description": "\\[c^{2}=a^{2}-b^{2}\\]\nIn any origin-centred ellipse, the three parameters relate like Pythagoras.",
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"fragment_index": 2,
"text_description": "Variable Definitions\na\nSemi-major axis length\nb\nSemi-minor axis length\nc\nFocus distance from centre",
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"text_description": "Applications\nConsistency Check\nTest numbers: does \\(4^{2}=5^{2}-3^{2}\\)? Yes, so they form an ellipse.\nLocate Foci\nCompute \\(c=\\sqrt{a^{2}-b^{2}}\\) then plot points \\((\\pm c,0)\\).",
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},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Visualising a, b, c",
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{
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"text_description": "Ellipse annotated with \\(2a, 2b\\) and \\(c\\).",
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{
"fragment_index": 2,
"text_description": "From Symbols to Lengths\nAn ellipse has three key lengths measured from its centre.\nSeeing them on the graph links symbols to real distances.",
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{
"fragment_index": 3,
"text_description": "Key Points:\n\\(2a\\): full major-axis length.\n\\(2b\\): full minor-axis length.\nFoci are \\(c\\) units from centre on the major axis.",
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},
{
"slide": 6,
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"text_description": "Standard Equation",
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"text_description": "1\n\\[PF_{1}+PF_{2}=2a\\]\nStart with the constant-sum definition of an ellipse.",
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{
"fragment_index": 2,
"text_description": "2\n\\[\\sqrt{(x+c)^{2}+y^{2}}+\\sqrt{(x-c)^{2}+y^{2}}=2a\\]\nExpress each distance using the distance formula.",
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"text_description": "3\n\\[\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\]\nSquare, simplify, and rearrange to obtain the standard form.",
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"fragment_index": 4,
"text_description": "Key Insight:\nAny \\((x,y)\\) that satisfies the equation lies on the ellipse.",
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]
},
{
"slide": 7,
"fragments": []
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{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Latus Rectum",
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"text_description": "Definition & Formula\nThe latus rectum is a special chord that passes through a focus and is perpendicular to the major axis.\nIts length is \\(2\\,\\frac{b^{2}}{a}\\).\nExample: with \\(a = 5\\) and \\(b = 3\\), length = \\(2\\,\\frac{3^{2}}{5} = \\frac{18}{5}\\) units.",
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"text_description": "Key Points:\nChord through a focus\nPerpendicular to the major axis\nLength \\(2b^{2}/a\\)",
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]
},
{
"slide": 9,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Takeaways\nEllipse essentials",
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{
"fragment_index": 1,
"text_description": "Definition\nEvery point keeps the sum of distances to two foci constant.",
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{
"fragment_index": 2,
"text_description": "Axes & Centre\nMajor axis length \\(2a\\) is longest, minor axis \\(2b\\) shortest; they intersect at the centre.",
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{
"fragment_index": 3,
"text_description": "Equation\nStandard form: \\(\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\).",
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{
"fragment_index": 4,
"text_description": "Parameter Link\nFocus distance obeys \\(c^{2}=a^{2}-b^{2}\\).",
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{
"fragment_index": 5,
"text_description": "Interactive Insight\nChanging \\(a\\) or \\(b\\) smoothly stretches or squeezes the curve.",
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]
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]