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[
{
"slide": 1,
"fragments": [
{
"fragment_index": -1,
"text_description": "Cone Slice View",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Tilted plane slicing a cone produces an ellipse.",
"image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/Xsw59cREc3RhrqIumCmFIAp5WfhmKaiN5foVhnT2.png"
},
{
"fragment_index": 2,
"text_description": "Ellipse from a Tilted Plane\nA right circular cone cut by a slanted plane, within a single nappe, creates an ellipse.\nMore tilt makes the ellipse slimmer; less tilt makes it closer to a circle.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key Points:\nPlane angle is steeper than the base but flatter than the cone side.\nCut stays within one nappe, giving a closed, oval curve.\nChanging the tilt alters size and eccentricity, not the conic type.",
"image_description": ""
}
]
},
{
"slide": 2,
"fragments": [
{
"fragment_index": -1,
"text_description": "Focus Magic",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Ellipse with foci \\(F_1\\) and \\(F_2\\)",
"image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/HwfCYJEMztL3zlpxxbxZMTJvwvqTz3vpdB6QfXxv.png"
},
{
"fragment_index": 2,
"text_description": "Constant-Sum Locus\nAn ellipse is the locus of all points \\(P\\) such that \\(PF_1 + PF_2\\) is constant.\nThat unchanging sum defines the curve’s size—move \\(P\\) anywhere and the value stays fixed.\nKey Points:\nFoci \\(F_1\\) and \\(F_2\\) stay fixed inside the ellipse.\nConstant sum \\(PF_1 + PF_2 = k\\).\nQuiz: If one measurement gives \\(10\\text{ cm}\\), what is \\(k\\) for all points?",
"image_description": ""
}
]
},
{
"slide": 3,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Distances\nApplications",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "\\[c^{2}=a^{2}-b^{2}\\]",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Variable Definitions\n\\(a\\)\nSemi-major axis length\n\\(b\\)\nSemi-minor axis length\n\\(c\\)\nFocal length (centre to focus)",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Find Foci\nGiven \\(a\\) and \\(b\\), compute \\(c\\) to locate the two foci.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Check Validity\nTest if lengths satisfy \\(c^{2}=a^{2}-b^{2}\\) to confirm an ellipse exists.",
"image_description": ""
}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": -1,
"text_description": "Equation Build-Up",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\n\\[\\sqrt{(x+c)^2+y^{2}}+\\sqrt{(x-c)^2+y^{2}}=2a\\]\nDistance-sum rule for an ellipse with foci \\((\\pm c,0)\\); major axis length \\(2a\\).",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\n\\[2x^{2}+2y^{2}+2c^{2}+2\\sqrt{\\bigl[(x+c)^2+y^{2}\\bigr]\\bigl[(x-c)^2+y^{2}\\bigr]}=4a^{2}\\]\nSquare once to begin eliminating radicals; collect like terms.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "3\n\\[b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}\\]\nIsolate the remaining radical, square again, and substitute \\(b^{2}=a^{2}-c^{2}\\).",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "4\n\\[\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\]\nDivide by \\(a^{2}b^{2}\\) to reach the standard equation centred at the origin.",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "Key Insight:\nThe link \\(b^{2}=a^{2}-c^{2}\\) ties focal spacing to axis lengths, sealing the derivation.",
"image_description": ""
}
]
},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Ellipse Plot",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "",
"image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/cZUrZ8Gi2pJlvdBylYpJv1DAgQcwPVRf01WP3b86.png"
},
{
"fragment_index": 2,
"text_description": "What the graph shows\nThis graphical view plots the ellipse \\( \\frac{x^{2}}{9} + \\frac{y^{2}}{4} = 1 \\) on the coordinate plane.\nThe curve crosses the x-axis at \\((\\pm3,0)\\) and the y-axis at \\((0,\\pm2)\\).\nThese intercepts equal the semi-major axis \\(a = 3\\) and semi-minor axis \\(b = 2\\) introduced earlier.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key Points:\nGraphical view makes the oval shape clear.\nEnds on x-axis: \\((\\pm3,0)\\).\nEnds on y-axis: \\((0,\\pm2)\\).",
"image_description": ""
}
]
},
{
"slide": 6,
"fragments": []
},
{
"slide": 7,
"fragments": [
{
"fragment_index": -1,
"text_description": "Test Yourself\nSubmit Answer\nCorrect!\nGreat job! You correctly used \\(b^{2}=a^{2}-c^{2}\\).\nIncorrect\nReview the Pythagorean-like relation for an ellipse: \\(b^{2}=a^{2}-c^{2}\\).\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 });",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Question\nFor an ellipse with semi-major axis \\(a\\) and focal distance \\(c\\), which formula gives the semi-minor axis \\(b\\)?",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "1\n\\(b^{2}=a^{2}+c^{2}\\)",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "2\n\\(b^{2}=a^{2}-c^{2}\\)",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "3\n\\(b^{2}=c^{2}-a^{2}\\)",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "4\n\\(b^{2}=2ac\\)",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Hint:\nThink of a right-angle triangle formed by \\(a, b,\\) and \\(c\\) at the centre of the ellipse.",
"image_description": ""
}
]
},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Ellipse Wrap-Up\nThank You!\nWe hope you found this lesson informative and engaging.",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Quick recap: an ellipse is the set of points whose summed distances to two foci is constant.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Standard form: \\( \\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1 \\).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Major axis length \\(2a\\); minor axis \\(2b\\); focal distance satisfies \\(c^2 = a^2 - b^2\\).",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Eccentricity \\( e=\\frac{c}{a} \\) gauges stretch; for every ellipse \\( 0 < e < 1 \\).",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "The larger denominator marks the major axis, revealing horizontal or vertical orientation.",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Translate or rotate the form to describe any ellipse you encounter in practice.",
"image_description": ""
}
]
}
]