{ "query": { "display": "asymptotes $$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1$$", "symbolab_question": "CONIC#asymptotes \\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "asymptotes", "default": "y=\\frac{3x}{4},\\quad y=-\\frac{3x}{4}" }, "steps": { "type": "interim", "title": "Hyperbola asymptotes given $$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1:{\\quad}y=\\frac{3x}{4},\\:\\quad\\:y=-\\frac{3x}{4}$$", "steps": [ { "type": "definition", "title": "Hyperbola asymptotes", "text": "The asymptotes are the lines the hyperbola tends to at $$\\pm\\infty$$<br/>For up-down hyperbola the asymptotes are $$y=\\pm\\frac{a}{b}\\left(x-h\\right)+k$$" }, { "type": "step", "result": "y=\\pm\\:\\frac{a}{b}\\left(x-h\\right)+k" }, { "type": "step", "primary": "Calculate Hyperbola properties" }, { "type": "interim", "title": "$$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=3,\\:b=4$$", "input": "\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1", "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": "step", "primary": "Rewrite $$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1\\:$$in the form of a standard hyperbola equation", "result": "\\frac{\\left(y-0\\right)^{2}}{3^{2}}-\\frac{\\left(x-0\\right)^{2}}{4^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=3,\\:b=4" } ], "meta": { "interimType": "Hyperbola UpDown Top Title 3Eq" } }, { "type": "step", "result": "y=\\frac{3}{4}\\left(x-0\\right)+0,\\:\\quad\\:y=-\\frac{3}{4}\\left(x-0\\right)+0" }, { "type": "step", "primary": "Refine", "result": "y=\\frac{3x}{4},\\:\\quad\\:y=-\\frac{3x}{4}" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{3x}{4}", "displayFormula": "y=\\frac{3x}{4}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{3x}{4}", "displayFormula": "y=-\\frac{3x}{4}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{9(\\frac{x^{2}}{4^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{3^{2}}-\\frac{x^{2}}{4^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{9(\\frac{x^{2}}{4^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{3^{2}}-\\frac{x^{2}}{4^{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": "3", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=3" ], "isAsymptote": false } }, { "p1x": "0", "p1y": "0", "p2x": "4", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=4" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{y^{2}}{3^{2}}-\\frac{x^{2}}{4^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -18, "xMax": 18, "yMin": -18, "yMax": 18 } }, "showViewLarger": true } } }