eccentricity 36x^2+y^2=36

{ "query": { "display": "eccentricity $$36x^{2}+y^{2}=36$$", "symbolab_question": "CONIC#eccentricity 36x^{2}+y^{2}=36" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Ellipse", "subTopic": "eccentricity", "default": "\\frac{\\sqrt{35}}{6}" }, "steps": { "type": "interim", "title": "Ellipse eccentricity given $$36x^{2}+y^{2}=36:{\\quad}\\frac{\\sqrt{35}}{6}$$", "steps": [ { "type": "definition", "title": "Ellipse eccentricity", "text": "The eccentricity is a measure of how much the ellipse deviates from a circle.<br/>For an ellipse with major axis parallel to the y-axis, the eccentricity is$$\\frac{\\sqrt{b^{2}-a^{2}}}{b}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{b^{2}-a^{2}}}{b}" }, { "type": "step", "primary": "Calculate ellipse properties" }, { "type": "interim", "title": "$$36x^{2}+y^{2}=36:{\\quad}$$Ellipse with center $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:b=6,\\:a=1$$", "input": "36x^{2}+y^{2}=36", "steps": [ { "type": "definition", "title": "Ellipse standard equation", "text": "$$\\frac{\\left(x-h\\right)^{2}}{a^2}+\\frac{\\left(y-k\\right)^{2}}{b^2}=1\\:$$is the ellipse standard equation<br/>with center $$\\left(h,\\:k\\right)\\:$$and $$a,\\:b$$ are the semi-major and semi-minor axes" }, { "type": "interim", "title": "Rewrite $$36x^{2}+y^{2}=36\\:$$in the form of the standard ellipse equation", "input": "36x^{2}+y^{2}=36", "steps": [ { "type": "step", "primary": "Divide by coefficient of square terms: $$36$$", "result": "x^{2}+\\frac{1}{36}y^{2}=1" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$1$$", "result": "\\frac{1}{1}x^{2}+\\frac{1}{36}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{x^{2}}{1}+\\frac{y^{2}}{36}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-0\\right)^{2}}{1^{2}}+\\frac{\\left(y-0\\right)^{2}}{6^{2}}=1" } ], "meta": { "interimType": "Ellipse Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(x-0\\right)^{2}}{1^{2}}+\\frac{\\left(y-0\\right)^{2}}{6^{2}}=1" }, { "type": "step", "primary": "Therefore ellipse properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=1,\\:b=6" }, { "type": "step", "primary": "$$b>a\\:$$therefore $$b\\:$$is semi-major axis and $$a\\:$$is semi-minor axis", "result": "\\mathrm{Ellipse\\:with\\:center}\\:\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:b=6,\\:a=1" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{\\sqrt{6^{2}-1^{2}}}{6}" }, { "type": "interim", "title": "$$\\frac{\\sqrt{6^{2}-1^{2}}}{6}=\\frac{\\sqrt{35}}{6}$$", "input": "\\frac{\\sqrt{6^{2}-1^{2}}}{6}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{\\sqrt{6^{2}-1}}{6}" }, { "type": "interim", "title": "$$\\sqrt{6^{2}-1}=\\sqrt{35}$$", "input": "\\sqrt{6^{2}-1}", "steps": [ { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36-1}" }, { "type": "step", "primary": "Subtract the numbers: $$36-1=35$$", "result": "=\\sqrt{35}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77KBNPXL7IUcKw9vy4OEq8nyRHuGw7+tM5METTDj6vVFCyVMMrE5H/on5k4a9Rxq6dWWdJyie4+0+O0C8JXGKp4RKNR+wXMiC9ZVIMS8aWNWybKKXnYvUXehSv0w06mf9HFX2jVGzZ70B521E5uGmKA==" } }, { "type": "step", "result": "=\\frac{\\sqrt{35}}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMdriXiO2BneadXoD3JOm6bYAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJuanHhSXYhLQB/goeZks8Imrys0VFr0ex+0yPqoQFyKdHDFhfCK1tNmWArYe/rItpNriXiO2BneadXoD3JOm6bayq6317hZP5NJRJD3bfH2/qkqc4qfL6ySo1C/0yz9uQQ==" } }, { "type": "step", "result": "=\\frac{\\sqrt{35}}{6}" } ], "meta": { "solvingClass": "Ellipse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { 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