-9x^2+y^2-72x-153=0

{ "query": { "display": "$$-9x^{2}+y^{2}-72x-153=0$$", "symbolab_question": "CONIC#-9x^{2}+y^{2}-72x-153=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "formula", "default": "(h,k)=(-4,0),a=3,b=1" }, "steps": { "type": "interim", "title": "$$-9x^{2}+y^{2}-72x-153=0:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(-4,\\:0\\right),\\:a=3,\\:b=1$$", "input": "-9x^{2}+y^{2}-72x-153=0", "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 $$-9x^{2}+y^{2}-72x-153=0\\:$$in the form of a standard hyperbola equation", "input": "-9x^{2}+y^{2}-72x-153=0", "steps": [ { "type": "step", "primary": "Add $$153$$ to both sides", "result": "-9x^{2}-72x+y^{2}=153" }, { "type": "step", "primary": "Factor out coefficient of square terms", "result": "-9\\left(x^{2}+8x\\right)+y^{2}=153" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$9$$", "result": "-\\left(x^{2}+8x\\right)+\\frac{1}{9}y^{2}=17" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$1$$", "result": "-\\frac{1}{1}\\left(x^{2}+8x\\right)+\\frac{1}{9}y^{2}=17" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "-\\frac{1}{1}\\left(x^{2}+8x+16\\right)+\\frac{1}{9}y^{2}=17-\\frac{1}{1}\\left(16\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "-\\frac{1}{1}\\left(x+4\\right)^{2}+\\frac{1}{9}y^{2}=17-\\frac{1}{1}\\left(16\\right)" }, { "type": "step", "primary": "Refine $$17-\\frac{1}{1}\\left(16\\right)$$", "result": "-\\frac{1}{1}\\left(x+4\\right)^{2}+\\frac{1}{9}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "-\\frac{\\left(x+4\\right)^{2}}{1}+\\frac{y^{2}}{9}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(y-0\\right)^{2}}{3^{2}}-\\frac{\\left(x-\\left(-4\\right)\\right)^{2}}{1^{2}}=1" } ], "meta": { "interimType": "Hyperbola Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(y-0\\right)^{2}}{3^{2}}-\\frac{\\left(x-\\left(-4\\right)\\right)^{2}}{1^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-4,\\:0\\right),\\:a=3,\\:b=1" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=3(x+4)", "displayFormula": "y=3(x+4)", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-3(x+4)", "displayFormula": "y=-3(x+4)", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{9(\\frac{(x-(-4))^{2}}{1^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{3^{2}}-\\frac{(x-(-4))^{2}}{1^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{9(\\frac{(x-(-4))^{2}}{1^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{3^{2}}-\\frac{(x-(-4))^{2}}{1^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-4,0)" ], "pointsDecimal": [ { "fst": -4, "snd": 0 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "-4", "p1y": "0", "p2x": "-4", "p2y": "3", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=3" ], "isAsymptote": false } }, { "p1x": "-4", "p1y": "0", "p2x": "-3", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=1" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{y^{2}}{3^{2}}-\\frac{(x-(-4))^{2}}{1^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -15.642857142857142, "xMax": 11.357142857142858, "yMin": -13.5, "yMax": 13.5 } }, "showViewLarger": true } } }