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