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[
{
"slide": 1,
"fragments": [
{
"fragment_index": 1,
"text_description": "What is an Ellipse?",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Ellipse",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "A locus of points in a plane whose sum of distances from two fixed points – the foci – is constant.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Quiz: If the constant sum is 10 cm, where can the foci never be? Apply the triangle inequality.",
"image_description": ""
}
]
},
{
"slide": 2,
"fragments": [
{
"fragment_index": -1,
"text_description": "Focus Magic",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Geometry sketch: ellipse with two foci \\(F_1\\) and \\(F_2\\).",
"image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/RhoWhn2aQHjfb6rZOocH3OcbsF2KKPaydGx899Sg.png"
},
{
"fragment_index": 2,
"text_description": "Constant-Sum Property\nEach blue point \\(P\\) on the sketch obeys \\(PF_1 + PF_2 = 10\\,\\text{cm}\\).\nThe fixed 10 cm string keeps its ends at the foci while the pointer roams, tracing the ellipse.\nKey Points:\nFoci \\(F_1\\) and \\(F_2\\) are highlighted in red.\nSum of distances to the foci stays constant at 10 cm.\nThis rule defines and visualises an ellipse.",
"image_description": ""
}
]
},
{
"slide": 3,
"fragments": [
{
"fragment_index": 1,
"text_description": "Standard Equation",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "\\[\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\]\nCentre at origin; major axis horizontal. Equation links coordinates to squared axis lengths \\(a^{2}\\) and \\(b^{2}\\).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Variable Definitions\na\nsemi-major axis (half horizontal length)\nb\nsemi-minor axis (half vertical length)",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Applications\nPlanetary orbit modelling\nPlanet paths around the Sun are nearly elliptical.\nOptical reflector design\nElliptical mirrors focus rays between two foci.",
"image_description": ""
}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": -1,
"text_description": "Focus Distance Relation",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\n\\[P(x, y)\\ \\text{ on }\\ \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\]\nAlgebraic model of the ellipse in standard form.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\n\\[\\lvert PF_{1}\\rvert=\\sqrt{(x-c)^{2}+y^{2}},\\quad \\lvert PF_{2}\\rvert=\\sqrt{(x+c)^{2}+y^{2}}\\]\nGeometry link: distances to foci \\(F_{1}(-c,0)\\) and \\(F_{2}(c,0)\\).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "3\n\\[\\lvert PF_{1}\\rvert+\\lvert PF_{2}\\rvert=2a \\;\\;\\Rightarrow\\;\\; c^{2}=a^{2}-b^{2}\\]\nSum property gives major axis length \\(2a\\); algebra proves \\(c^{2}=a^{2}-b^{2}\\).",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Key Insight:\nAs \\(b\\) approaches \\(a\\), the ellipse flattens into a circle and the foci converge to the centre.",
"image_description": ""
}
]
},
{
"slide": 5,
"fragments": []
},
{
"slide": 6,
"fragments": [
{
"fragment_index": -1,
"text_description": "Eccentricity Unpacked",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\n🧮\nDefine\nEccentricity \\(e=\\frac{c}{a}\\). For any ellipse \\(0\\le e<1\\).",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\n🔍\nSpecial cases\n\\(e=0\\) gives a circle; \\(e\\) near 1 yields a stretched ellipse.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "3\n📏\nCompute quickly\nFind \\(c=\\sqrt{a^{2}-b^{2}}\\) then \\(e=\\frac{c}{a}\\).",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Pro Tip:\nLarger \\(e\\) signals more flattening.",
"image_description": ""
}
]
},
{
"slide": 7,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nQuestion\nWhich equation is an ellipse centred at the origin with its major axis along the \\(y\\)-axis?\nSubmit Answer\nCorrect!\nCorrect—larger denominator under y^2 means vertical major axis.\nIncorrect\nNot quite. Remember: ellipse needs both squared terms positive and added.\nconst correctOption = 0;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('slide-08-a1b2c3-submitBtn');\n const feedbackCorrect = document.getElementById('slide-08-a1b2c3-feedback-correct');\n const feedbackIncorrect = document.getElementById('slide-08-a1b2c3-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 });",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\n\\( \\dfrac{x^{2}}{9} + \\dfrac{y^{2}}{25} = 1 \\)",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\n\\( \\dfrac{x^{2}}{16} - \\dfrac{y^{2}}{9} = 1 \\)",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "3\n\\( y = 2x^{2} + 3 \\)",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "4\n\\( (x-2)^{2} + (y+1)^{2} = 16 \\)",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "Hint:\nLook for the plus sign and unequal positive denominators.",
"image_description": ""
}
]
},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Takeaways\nEllipses in a nutshell",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "🎯\nLocus idea\nSum of distances to two foci stays constant.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "📐\nStandard form\n\\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 \\)\\, (\\(a \\ge b\\)).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "🔗\nFoci distance\n\\( c^{2} = a^{2} - b^{2} \\) links axes to foci.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "📊\nEccentricity\n\\( e = \\frac{c}{a} \\) quantifies ovalness.",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "🛠️\nDynamic view\nChanging \\( a \\) & \\( b \\) reshapes, keeping area \\( \\pi a b \\).",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "🚀\nNext\nUp next: parametric form & real-world applications.",
"image_description": ""
}
]
}
]