9x^2-y^2-36x-6y+18=0

{ "query": { "display": "$$9x^{2}-y^{2}-36x-6y+18=0$$", "symbolab_question": "CONIC#9x^{2}-y^{2}-36x-6y+18=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "formula", "default": "(h,k)=(2,-3),a=1,b=3" }, "steps": { "type": "interim", "title": "$$9x^{2}-y^{2}-36x-6y+18=0:\\quad$$Right-left Hyperbola with $$\\left(h,\\:k\\right)=\\left(2,\\:-3\\right),\\:a=1,\\:b=3$$", "input": "9x^{2}-y^{2}-36x-6y+18=0", "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": "interim", "title": "Rewrite $$9x^{2}-y^{2}-36x-6y+18=0\\:$$in the form of a standard hyperbola equation", "input": "9x^{2}-y^{2}-36x-6y+18=0", "steps": [ { "type": "step", "primary": "Subtract $$18$$ from both sides", "result": "9x^{2}-36x-y^{2}-6y=-18" }, { "type": "step", "primary": "Factor out coefficient of square terms", "result": "9\\left(x^{2}-4x\\right)-\\left(y^{2}+6y\\right)=-18" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$9$$", "result": "\\left(x^{2}-4x\\right)-\\frac{1}{9}\\left(y^{2}+6y\\right)=-2" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$1$$", "result": "\\frac{1}{1}\\left(x^{2}-4x\\right)-\\frac{1}{9}\\left(y^{2}+6y\\right)=-2" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\frac{1}{1}\\left(x^{2}-4x+4\\right)-\\frac{1}{9}\\left(y^{2}+6y\\right)=-2+\\frac{1}{1}\\left(4\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "\\frac{1}{1}\\left(x-2\\right)^{2}-\\frac{1}{9}\\left(y^{2}+6y\\right)=-2+\\frac{1}{1}\\left(4\\right)" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\frac{1}{1}\\left(x-2\\right)^{2}-\\frac{1}{9}\\left(y^{2}+6y+9\\right)=-2+\\frac{1}{1}\\left(4\\right)-\\frac{1}{9}\\left(9\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "\\frac{1}{1}\\left(x-2\\right)^{2}-\\frac{1}{9}\\left(y+3\\right)^{2}=-2+\\frac{1}{1}\\left(4\\right)-\\frac{1}{9}\\left(9\\right)" }, { "type": "step", "primary": "Refine $$-2+\\frac{1}{1}\\left(4\\right)-\\frac{1}{9}\\left(9\\right)$$", "result": "\\frac{1}{1}\\left(x-2\\right)^{2}-\\frac{1}{9}\\left(y+3\\right)^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{\\left(x-2\\right)^{2}}{1}-\\frac{\\left(y+3\\right)^{2}}{9}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-2\\right)^{2}}{1^{2}}-\\frac{\\left(y-\\left(-3\\right)\\right)^{2}}{3^{2}}=1" } ], "meta": { "interimType": "Hyperbola Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(x-2\\right)^{2}}{1^{2}}-\\frac{\\left(y-\\left(-3\\right)\\right)^{2}}{3^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(2,\\:-3\\right),\\:a=1,\\:b=3" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=3(x-2)-3", "displayFormula": "y=3(x-2)-3", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-3(x-2)-3", "displayFormula": "y=-3(x-2)-3", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{9(\\frac{(x-2)^{2}}{1^{2}}-1)}-3", "displayFormula": "\\frac{(x-2)^{2}}{1^{2}}-\\frac{(y-(-3))^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{9(\\frac{(x-2)^{2}}{1^{2}}-1)}-3", "displayFormula": "\\frac{(x-2)^{2}}{1^{2}}-\\frac{(y-(-3))^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(2,-3)" ], "pointsDecimal": [ { "fst": 2, "snd": -3 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "2", "p1y": "-3", "p2x": "3", "p2y": "-3", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=1" ], "isAsymptote": false } }, { "p1x": "2", "p1y": "-3", "p2x": "2", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=3" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{(x-2)^{2}}{1^{2}}-\\frac{(y-(-3))^{2}}{3^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -12.214285714285714, "xMax": 14.785714285714286, "yMin": -16.02857142857143, "yMax": 10.971428571428572 } }, "showViewLarger": true } } }