{ "query": { "display": "vertices $$\\frac{x^{2}}{25}-\\frac{y^{2}}{9}=1$$", "symbolab_question": "CONIC#vertices \\frac{x^{2}}{25}-\\frac{y^{2}}{9}=1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "vertices", "default": "(5,0),(-5,0)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Hyperbola vertices given $$\\frac{x^{2}}{25}-\\frac{y^{2}}{9}=1:{\\quad}\\left(5,\\:0\\right),\\:\\left(-5,\\:0\\right)$$", "steps": [ { "type": "definition", "title": "Hyperbola vertices", "text": "The vertices $$\\left(h+a,\\:k\\right),\\:\\left(h-a,\\:k\\right)\\:$$are the two bending points of the hyperbola with center $$\\left(h,\\:k\\right)\\:$$and semi-axis $$a,\\:b.\\:$$" }, { "type": "step", "result": "\\left(h+a,\\:k\\right),\\:\\left(h-a,\\:k\\right)" }, { "type": "step", "primary": "Calculate Hyperbola properties" }, { "type": "interim", "title": "$$\\frac{x^{2}}{25}-\\frac{y^{2}}{9}=1:\\quad$$Right-left Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=3$$", "input": "\\frac{x^{2}}{25}-\\frac{y^{2}}{9}=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 $$\\frac{x^{2}}{25}-\\frac{y^{2}}{9}=1\\:$$in the form of a standard hyperbola equation", "result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}-\\frac{\\left(y-0\\right)^{2}}{3^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=3" } ], "meta": { "interimType": "Hyperbola RightLeft Top Title 3Eq" } }, { "type": "step", "result": "\\left(0+5,\\:0\\right),\\:\\left(0-5,\\:0\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(5,\\:0\\right),\\:\\left(-5,\\:0\\right)" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{3x}{5}", "displayFormula": "y=\\frac{3x}{5}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{3x}{5}", "displayFormula": "y=-\\frac{3x}{5}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{9(\\frac{x^{2}}{5^{2}}-1)}", "displayFormula": "\\frac{x^{2}}{5^{2}}-\\frac{y^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{9(\\frac{x^{2}}{5^{2}}-1)}", "displayFormula": "\\frac{x^{2}}{5^{2}}-\\frac{y^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(5,0)", "(-5,0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 5, "snd": 0 }, { "fst": -5, "snd": 0 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "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": "3", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=3" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{x^{2}}{5^{2}}-\\frac{y^{2}}{3^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -22.5, "xMax": 22.5, "yMin": -22.5, "yMax": 22.5 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }