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[
{
"slide": 1,
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"fragment_index": -1,
"text_description": "Diving into Ellipses\nTracing perfect ovals from chalkboards to cosmic orbits.",
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"slide": 2,
"fragments": [
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"fragment_index": -1,
"text_description": "What is an Ellipse?",
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"text_description": "Fig 10.20 | Ellipse with foci \\(F_1,F_2\\); blue segments keep constant sum.",
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"text_description": "Formal Definition (Definition 4)\nDefinition 4: An ellipse is the locus of points whose distances to two foci add to a constant.\nThus, for every point \\(P\\), \\(PF_1 + PF_2 = 2a\\), a fixed length greater than \\(F_1F_2\\).\nKey Points:\n\\(F_1\\) and \\(F_2\\) are the fixed foci.\nBlue segments in Fig 10.20 mark \\(PF_1\\) and \\(PF_2\\).\nTheir sum stays constant (\\(2a\\)), keeping \\(P\\) on the ellipse.",
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{
"slide": 3,
"fragments": [
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"fragment_index": -1,
"text_description": "Key Parts of an Ellipse\nCentre, Axes & Vertices\nLook at Fig 10.21 and match each label. After this, you should name the centre, vertices, major and minor axes with ease.",
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"fragment_index": 1,
"text_description": "Fig 10.21 — Ellipse showing major and minor axes",
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"text_description": "Key Points:\nMajor axis — longest line through both foci.\nMinor axis — shortest line, perpendicular to major axis at the centre.\nVertices — two endpoints of the major axis.\nCentre — midpoint of the line joining the foci.",
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{
"slide": 4,
"fragments": [
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"fragment_index": -1,
"text_description": "Meet a, b and c",
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"fragment_index": 2,
"text_description": "Parameters of an Ellipse\nIn Fig 10.22 and Fig 10.23, three numbers fully describe any ellipse drawn with centre O.\nKnow them, and you can link the semi-axes to the focus distance.",
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{
"fragment_index": 3,
"text_description": "Key Points:\n\\(a\\): semi-major axis\n\\(b\\): semi-minor axis\n\\(c\\): centre \\(\\rightarrow\\) focus distance\nRelationship: \\(a^{2}=b^{2}+c^{2}\\)",
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},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Proving a² = b² + c²",
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"fragment_index": 1,
"text_description": "1\n\\[PF_1+PF_2 = 2a\\]\nPlace P on the major axis; ellipse definition gives the sum as the full length \\(2a\\).",
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{
"fragment_index": 2,
"text_description": "2\n\\[QF_1+QF_2 = 2\\sqrt{b^{2}+c^{2}}\\]\nPick Q on the minor axis; each focal distance forms a right triangle, giving \\(\\sqrt{b^{2}+c^{2}}\\).",
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"fragment_index": 3,
"text_description": "3\n\\[2a = 2\\sqrt{b^{2}+c^{2}}\\]\nSince the sum is constant for every point, equate results from P and Q.",
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"fragment_index": 4,
"text_description": "4\n\\[a^{2}=b^{2}+c^{2}\\]\nSquare both sides to relate semi-major, semi-minor, and focal lengths.",
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{
"fragment_index": 5,
"text_description": "Key Insight:\nComparing sums at symmetric points links the ellipse’s constant distance property directly to \\(a\\), \\(b\\), and \\(c\\).",
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]
},
{
"slide": 6,
"fragments": []
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{
"slide": 7,
"fragments": [
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"fragment_index": 1,
"text_description": "Standard Equations",
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{
"fragment_index": 2,
"text_description": "Fig 10.24 • Ellipse with centre at O(0,0)",
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"fragment_index": 3,
"text_description": "Centre at (0, 0)",
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{
"fragment_index": 4,
"text_description": "A centred ellipse has two orientations decided by its major axis.",
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{
"fragment_index": 5,
"text_description": "Here \\(a\\) is the semi-major length and \\(b\\) the semi-minor, with \\(a > b\\).",
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"fragment_index": 6,
"text_description": "Key Points:\nMajor axis on x-axis: \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 \\)\nMajor axis on y-axis: \\( \\frac{x^{2}}{b^{2}} + \\frac{y^{2}}{a^{2}} = 1 \\)",
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},
{
"slide": 8,
"fragments": [
{
"fragment_index": 1,
"text_description": "Deriving x²/a² + y²/b² = 1",
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{
"fragment_index": 2,
"text_description": "Fig 10.25 – Point P on an ellipse with foci \\(F_1,F_2\\)",
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"fragment_index": 3,
"text_description": "From Distance Rule to Standard Equation",
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"fragment_index": 4,
"text_description": "Fig 10.25 places \\(P(x,y)\\) on an ellipse whose foci are \\(F_1(-c,0)\\) and \\(F_2(c,0)\\).",
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{
"fragment_index": 5,
"text_description": "By definition, the sum of distances satisfies \\(PF_1 + PF_2 = 2a\\).",
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"fragment_index": 6,
"text_description": "Key Steps:\nSubstitute \\(PF_1=\\sqrt{(x+c)^2+y^2}\\) and \\(PF_2=\\sqrt{(x-c)^2+y^2}\\).\nSquare once to remove one root; rearrange terms.\nSquare again to clear remaining radical.\nUse \\(b^{2}=a^{2}-c^{2}\\) to obtain \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\).",
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]
},
{
"slide": 9,
"fragments": [
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"fragment_index": -1,
"text_description": "Quick Properties",
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{
"fragment_index": 1,
"text_description": "Main Points\n1\nSymmetric about both x- and y-axes.\n2\nFoci lie on the major axis, equidistant from the centre.\n3\nLarger denominator ⇒ major axis in \\( \\dfrac{x^{2}}{a^{2}} + \\dfrac{y^{2}}{b^{2}} = 1 \\).",
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},
{
"fragment_index": 2,
"text_description": "Key Highlights\nSketch one quadrant; mirror using symmetry.\nLocate foci at \\((\\pm c,0)\\) or \\((0,\\pm c)\\) along the major axis.\nIdentify the major axis first; other features follow.",
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]
},
{
"slide": 10,
"fragments": [
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"fragment_index": 1,
"text_description": "Latus Rectum",
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"text_description": "",
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"fragment_index": 3,
"text_description": "Chord Through the Focus\nThe latus rectum is the chord that passes through a focus and is perpendicular to the major axis.\nKnowing its length lets us quickly compare the “width” of different ellipses at their foci.",
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{
"fragment_index": 4,
"text_description": "Key Points:\nLength \\(= 2\\frac{b^{2}}{a}\\)\nUses semi-major axis \\(a\\) and semi-minor axis \\(b\\).\nReferenced in Sec. 10.5.4 and Fig 10.26.",
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]
},
{
"slide": 11,
"fragments": [
{
"fragment_index": -1,
"text_description": "Worked Example 1\nExample 9 — Find foci, vertices, eccentricity and latus rectum of \\( \\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1 \\).",
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{
"fragment_index": 1,
"text_description": "1\nIdentify semi-axes\nCompare with \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) to get \\(a=5,\\,b=3\\).",
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"fragment_index": 2,
"text_description": "2\nCompute focal distance\n\\(c^{2}=a^{2}-b^{2}=25-9=16 \\Rightarrow c=4\\).",
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"fragment_index": 3,
"text_description": "3\nLocate foci & vertices\nMajor axis lies on \\(x\\)-axis ⇒ foci at \\((\\pm4,0)\\); vertices at \\((\\pm5,0)\\).",
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"fragment_index": 4,
"text_description": "4\nEccentricity & latus rectum\n\\(e=\\frac{c}{a}=\\frac{4}{5}=0.8\\); length of latus rectum \\(=\\frac{2b^{2}}{a}=\\frac{18}{5}\\).",
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},
{
"fragment_index": 5,
"text_description": "Pro Tip:\nFor any ellipse centred at the origin, remember \\(c^{2}=a^{2}-b^{2}\\).",
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}
]
},
{
"slide": 12,
"fragments": [
{
"fragment_index": -1,
"text_description": "Horizontal vs Vertical",
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{
"fragment_index": 1,
"text_description": "Horizontal Ellipse\n\\( \\dfrac{(x-h)^2}{a^2} + \\dfrac{(y-k)^2}{b^2} = 1,\\; a>b \\)\nMajor axis \\(2a\\) runs along the x-axis.\nFoci \\((h\\pm c,\\,k)\\), \\( c=\\sqrt{a^2-b^2} \\).\nExample 10: \\( \\dfrac{x^{2}}{25} + \\dfrac{y^{2}}{9}=1 \\).",
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{
"fragment_index": 2,
"text_description": "Vertical Ellipse\n\\( \\dfrac{(x-h)^2}{b^2} + \\dfrac{(y-k)^2}{a^2} = 1,\\; a>b \\)\nMajor axis \\(2a\\) runs along the y-axis.\nFoci \\((h,\\,k\\pm c)\\).\nExample 10: \\( \\dfrac{x^{2}}{9} + \\dfrac{y^{2}}{25}=1 \\).",
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{
"fragment_index": 3,
"text_description": "Key Similarities\nSame centre \\((h,k)\\).\nRelation \\(c^2 = a^2 - b^2\\) holds.\nEccentricity \\(e=\\dfrac{c}{a}\\) identical.\nLatus rectum length \\( \\dfrac{2b^2}{a} \\) same.",
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]
},
{
"slide": 13,
"fragments": [
{
"fragment_index": -1,
"text_description": "Find that Equation\nUse vertices & foci to model each ellipse.",
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"fragment_index": 1,
"text_description": "1\nExample 11 – horizontal axis\nVertices at \\((\\pm13,0)\\) give \\(a=13\\). Foci at \\((\\pm5,0)\\) give \\(c=5\\). Then \\(b=\\sqrt{a^{2}-c^{2}}=\\sqrt{169-25}=12\\). Equation: \\(\\dfrac{x^{2}}{169}+\\dfrac{y^{2}}{144}=1\\).",
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"fragment_index": 2,
"text_description": "2\nExample 12 – vertical axis\nMajor axis length \\(20\\Rightarrow a=10\\). Foci distance \\(5\\Rightarrow c=5\\). Hence \\(b^{2}=a^{2}-c^{2}=100-25=75\\). Equation: \\(\\dfrac{x^{2}}{75}+\\dfrac{y^{2}}{100}=1\\).",
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{
"fragment_index": 3,
"text_description": "Pro Tip:\nRemember: \\(b^{2}=a^{2}-c^{2}\\) for every ellipse — your shortcut to quick equations.",
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}
]
},
{
"slide": 14,
"fragments": [
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"fragment_index": -1,
"text_description": "Multiple Choice Question\nSubmit Answer\nCorrect!\nWell done! The larger denominator 25 under \\(y^{2}\\) confirms a vertical major axis.\nIncorrect\nRemember: the ellipse stretches along the axis whose denominator is larger. Try again!\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\nIdentify the standard form of an ellipse whose major axis lies on the y-axis.",
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"fragment_index": 2,
"text_description": "1\n\\( \\frac{x^{2}}{9} + \\frac{y^{2}}{4} = 1 \\)",
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},
{
"fragment_index": 3,
"text_description": "2\n\\( \\frac{x^{2}}{4} + \\frac{y^{2}}{25} = 1 \\)",
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},
{
"fragment_index": 4,
"text_description": "3\n\\( \\frac{x^{2}}{16} + \\frac{y^{2}}{9} = 1 \\)",
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{
"fragment_index": 5,
"text_description": "Hint:\nThe larger denominator shows the major axis. In Exercise 10.3 intro we learnt: if it is under \\(y^{2}\\), the axis is vertical.",
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"slide": 15,
"fragments": [
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"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": "An ellipse is the set of points whose distances to two foci sum to a constant.",
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{
"fragment_index": 2,
"text_description": "Semi-major \\(a\\), semi-minor \\(b\\), focal length \\(c\\) satisfy \\(c^{2}=a^{2}-b^{2}\\).",
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"fragment_index": 3,
"text_description": "Standard forms: \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) and \\( \\frac{x^{2}}{b^{2}}+\\frac{y^{2}}{a^{2}}=1 \\).",
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"text_description": "Eccentricity \\(e=\\frac{c}{a}\\); length of latus rectum \\( \\frac{2b^{2}}{a} \\).",
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"fragment_index": 5,
"text_description": "Use solved examples, then attempt the practice set to consolidate learning.",
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{
"fragment_index": 6,
"text_description": "Next Steps\nReview the summary, finish the practice set, and bring questions to the next class.",
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