4x^2+25y^2=100

{ "query": { "display": "$$4x^{2}+25y^{2}=100$$", "symbolab_question": "CONIC#4x^{2}+25y^{2}=100" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Ellipse", "subTopic": "formula", "default": "(h,k)=(0,0),a=5,b=2" }, "steps": { "type": "interim", "title": "$$4x^{2}+25y^{2}=100:{\\quad}$$Ellipse with center $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=2$$", "input": "4x^{2}+25y^{2}=100", "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 $$4x^{2}+25y^{2}=100\\:$$in the form of the standard ellipse equation", "input": "4x^{2}+25y^{2}=100", "steps": [ { "type": "step", "primary": "Divide by coefficient of square terms: $$4$$", "result": "x^{2}+\\frac{25}{4}y^{2}=25" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$25$$", "result": "\\frac{1}{25}x^{2}+\\frac{1}{4}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{x^{2}}{25}+\\frac{y^{2}}{4}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}+\\frac{\\left(y-0\\right)^{2}}{2^{2}}=1" } ], "meta": { "interimType": "Ellipse Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}+\\frac{\\left(y-0\\right)^{2}}{2^{2}}=1" }, { "type": "step", "primary": "Therefore ellipse properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=2" }, { "type": "step", "primary": "$$a>b\\:$$therefore $$a\\:$$is semi-major axis and $$b\\:$$is semi-minor axis", "result": "\\mathrm{Ellipse\\:with\\:center}\\:\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=2" } ], "meta": { "solvingClass": "Ellipse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(1-\\frac{x^{2}}{5^{2}})}", "displayFormula": "\\frac{x^{2}}{5^{2}}+\\frac{y^{2}}{2^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(1-\\frac{x^{2}}{5^{2}})}", "displayFormula": "\\frac{x^{2}}{5^{2}}+\\frac{y^{2}}{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": "5", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=5" ], "isAsymptote": false } }, { "p1x": "0", "p1y": "0", "p2x": "0", "p2y": "2", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=2" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{x^{2}}{5^{2}}+\\frac{y^{2}}{2^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -11.25, "xMax": 11.25, "yMin": -11.25, "yMax": 11.25 } }, "showViewLarger": true } } }