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[
{
"slide": 1,
"fragments": [
{
"fragment_index": -1,
"text_description": "What is an Ellipse?",
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},
{
"fragment_index": 1,
"text_description": "Ellipse",
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{
"fragment_index": 2,
"text_description": "The locus of plane points whose distances to two fixed points (foci) add up to the same constant.\nThis is the formal locus definition of an ellipse.",
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]
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{
"slide": 2,
"fragments": [
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"fragment_index": -1,
"text_description": "Standard Equation View",
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"fragment_index": 1,
"text_description": "",
"image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/7SZhv7vips4TPZdOkbwzkNTfxsJ6FARgtOVLO4Cf.png"
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"fragment_index": 2,
"text_description": "Example: \\( \\frac{x^{2}}{9} + \\frac{y^{2}}{4} = 1 \\)\nEvery point \\((x, y)\\) on this curve obeys the standard equation.\nSemi-major axis: \\(a = 3\\) units along the x-direction.\nSemi-minor axis: \\(b = 2\\) units along the y-direction, giving a tighter vertical span.\nKey Points:\nStandard form links algebra to shape: \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 \\).\nSemi-major axis \\(a = 3\\) → wider spread along x-axis.\nSemi-minor axis \\(b = 2\\) → narrower stretch along y-axis.",
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{
"slide": 3,
"fragments": [
{
"fragment_index": -1,
"text_description": "Focus–Directrix Insight\nOne focus and one line can generate every ellipse—here’s how.",
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{
"fragment_index": 1,
"text_description": "1\nPick Focus & Directrix\nFix a point \\(F\\) and a non-intersecting line \\(D\\) in the plane.",
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{
"fragment_index": 2,
"text_description": "2\nSet Eccentricity \\(e\\)\nChoose \\(0 < e < 1\\) and demand the ratio \\(\\\\frac{PF}{PD}=e\\) for all positions of \\(P\\).",
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{
"fragment_index": 3,
"text_description": "3\nTrace the Locus\nThe set of all points \\(P\\) satisfying the ratio forms a smooth closed curve—the ellipse.",
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{
"fragment_index": 4,
"text_description": "Pro Tip:\nSmaller \\(e\\) makes the ellipse more circular; as \\(e\\) approaches 1, it stretches out.",
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}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": 1,
"text_description": "Equation Derivation",
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{
"fragment_index": 2,
"text_description": "1\n\\[PF + PF' = 2a\\]\nBy definition, an ellipse keeps the sum of distances from any point \\(P\\) to the foci constant \\(2a\\).",
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{
"fragment_index": 3,
"text_description": "2\n\\[\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a\\]\nInsert coordinates \\(F(-c,0)\\), \\(F'(c,0)\\) and apply the distance formula to point \\(P(x,y)\\).",
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{
"fragment_index": 4,
"text_description": "3\n\\[\\sqrt{(x+c)^2 + y^2} = 2a - \\sqrt{(x-c)^2 + y^2}\\]\nIsolate one radical to prepare for systematic squaring.",
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},
{
"fragment_index": 5,
"text_description": "4\n\\[(x+c)^2 + y^2 = 4a^2 - 4a\\sqrt{(x-c)^2 + y^2} + (x-c)^2 + y^2\\]\nSquare both sides once, expand, and collect like terms.",
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{
"fragment_index": 6,
"text_description": "5\n\\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1,\\; b^2 = a^2 - c^2\\]\nAfter eliminating radicals and simplifying, divide by \\(a^2b^2\\) to obtain the standard form.",
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},
{
"fragment_index": 7,
"text_description": "Key Insight:\nGeometry meets algebra through \\(b^2 = a^2 - c^2\\); this link turns the distance-sum rule into the elegant equation \\(x^2/a^2 + y^2/b^2 = 1\\).",
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]
},
{
"slide": 5,
"fragments": [
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"fragment_index": -1,
"text_description": "Multiple Choice Question\nQuestion\nWhich statement is true for every point on an ellipse with foci \\(F_{1}\\) and \\(F_{2}\\)?\nHint:\nRecall the geometric definition of an ellipse.\nSubmit Answer\nCorrect!\nExactly. A fixed sum of distances defines an ellipse.\nIncorrect\nCheck the definition: for an ellipse, the sum (not difference or product) stays constant.\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": "1\nDistance to \\(F_{1}\\) equals distance to \\(F_{2}\\).",
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{
"fragment_index": 2,
"text_description": "2\nSum of distances to \\(F_{1}\\) and \\(F_{2}\\) is constant.",
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},
{
"fragment_index": 3,
"text_description": "3\nDifference of distances to \\(F_{1}\\) and \\(F_{2}\\) is constant.",
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{
"fragment_index": 4,
"text_description": "4\nProduct of distances to \\(F_{1}\\) and \\(F_{2}\\) is constant.",
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}
]
},
{
"slide": 6,
"fragments": []
},
{
"slide": 7,
"fragments": [
{
"fragment_index": -1,
"text_description": "Ellipse vs Circle",
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{
"fragment_index": 1,
"text_description": "Ellipse\nSum of distances to two foci is constant.\nAxes lengths unequal, \\(a \\neq b\\).\nEccentricity \\(0 < e < 1\\) → shows stretch.\nEquation \\(\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\).",
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{
"fragment_index": 2,
"text_description": "Circle\nAll points at equal distance from one centre.\nAxes lengths equal, \\(a = b\\).\nEccentricity \\(e = 0\\) → no stretch.\nEquation \\(x^{2} + y^{2} = r^{2}\\).",
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{
"fragment_index": 3,
"text_description": "Key Similarities\nClosed conic sections from slicing a right circular cone.\nSymmetric about both the major and minor axes.\nRepresented by second-degree equations in \\(x\\) and \\(y\\).",
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},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Takeaways",
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"fragment_index": 1,
"text_description": "Foci Anchor Shape\nAn ellipse is all points whose distances to two foci add to a constant.",
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},
{
"fragment_index": 2,
"text_description": "Equation Reveals Geometry\nStandard form \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}} = 1 \\) turns geometric distances into algebraic balance.",
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{
"fragment_index": 3,
"text_description": "Parameters Shape Size\nSemi-axes \\(a, b\\) set width and height; eccentricity \\(e=\\sqrt{1-\\frac{b^{2}}{a^{2}}}\\) gauges flatness.",
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}
]
}
]