9y^2-16x^2=144

{ "query": { "display": "$$9y^{2}-16x^{2}=144$$", "symbolab_question": "CONIC#9y^{2}-16x^{2}=144" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "formula", "default": "(h,k)=(0,0),a=4,b=3" }, "steps": { "type": "interim", "title": "$$9y^{2}-16x^{2}=144:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=4,\\:b=3$$", "input": "9y^{2}-16x^{2}=144", "steps": [ { "type": "definition", "title": "Hyperbola standard equation", "text": "$$\\frac{\\left(y-k\\right)^{2}}{a^2}-\\frac{\\left(x-h\\right)^{2}}{b^2}=1\\:$$ is the standard equation for an up-down facing hyperbola<br/>with center $$\\bold{\\left(h,\\:k\\right)},\\:$$ semi-axis $$\\bold{a}$$ and semi-conjugate-axis $$\\bold{b}$$." }, { "type": "interim", "title": "Rewrite $$9y^{2}-16x^{2}=144\\:$$in the form of a standard hyperbola equation", "input": "9y^{2}-16x^{2}=144", "steps": [ { "type": "step", "primary": "Divide by coefficient of square terms: $$16$$", "result": "-x^{2}+\\frac{9}{16}y^{2}=9" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$9$$", "result": "-\\frac{1}{9}x^{2}+\\frac{1}{16}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "-\\frac{x^{2}}{9}+\\frac{y^{2}}{16}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(y-0\\right)^{2}}{4^{2}}-\\frac{\\left(x-0\\right)^{2}}{3^{2}}=1" } ], "meta": { "interimType": "Hyperbola Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(y-0\\right)^{2}}{4^{2}}-\\frac{\\left(x-0\\right)^{2}}{3^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=4,\\:b=3" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{4x}{3}", "displayFormula": "y=\\frac{4x}{3}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{4x}{3}", "displayFormula": "y=-\\frac{4x}{3}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{16(\\frac{x^{2}}{3^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{4^{2}}-\\frac{x^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{16(\\frac{x^{2}}{3^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{4^{2}}-\\frac{x^{2}}{3^{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": "0", "p2y": "4", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=4" ], "isAsymptote": false } }, { "p1x": "0", "p1y": "0", "p2x": "3", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=3" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{y^{2}}{4^{2}}-\\frac{x^{2}}{3^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -18, "xMax": 18, "yMin": -18, "yMax": 18 } }, "showViewLarger": true } } }