2x^2+y^2=2

{ "query": { "display": "$$2x^{2}+y^{2}=2$$", "symbolab_question": "CONIC#2x^{2}+y^{2}=2" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Ellipse", "subTopic": "formula", "default": "(h,k)=(0,0),a=1,b=\\sqrt{2}" }, "steps": { "type": "interim", "title": "$$2x^{2}+y^{2}=2:{\\quad}$$Ellipse with center $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:b=\\sqrt{2},\\:a=1$$", "input": "2x^{2}+y^{2}=2", "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 $$2x^{2}+y^{2}=2\\:$$in the form of the standard ellipse equation", "input": "2x^{2}+y^{2}=2", "steps": [ { "type": "step", "primary": "Divide by coefficient of square terms: $$2$$", "result": "x^{2}+\\frac{1}{2}y^{2}=1" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$1$$", "result": "\\frac{1}{1}x^{2}+\\frac{1}{2}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{x^{2}}{1}+\\frac{y^{2}}{2}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-0\\right)^{2}}{1^{2}}+\\frac{\\left(y-0\\right)^{2}}{\\left(\\sqrt{2}\\right)^{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}}{\\left(\\sqrt{2}\\right)^{2}}=1" }, { "type": "step", "primary": "Therefore ellipse properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=1,\\:b=\\sqrt{2}" }, { "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=\\sqrt{2},\\:a=1" } ], "meta": { "solvingClass": "Ellipse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{2(1-\\frac{x^{2}}{1^{2}})}", "displayFormula": "\\frac{x^{2}}{1^{2}}+\\frac{y^{2}}{\\sqrt{2}^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{2(1-\\frac{x^{2}}{1^{2}})}", "displayFormula": "\\frac{x^{2}}{1^{2}}+\\frac{y^{2}}{\\sqrt{2}^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "0", "p1y": "0", "p2x": "1", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=1" ], "isAsymptote": false } }, { "p1x": "0", "p1y": "0", "p2x": "0", "p2y": "\\sqrt{2}", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=\\sqrt{2}" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{x^{2}}{1^{2}}+\\frac{y^{2}}{(\\sqrt{2})^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -3.1819805153394642, "xMax": 3.1819805153394642, "yMin": -3.1819805153394642, "yMax": 3.1819805153394642 } }, "showViewLarger": true } } }