x^2-2x-4y+9=0

{ "query": { "display": "$$x^{2}-2x-4y+9=0$$", "symbolab_question": "CONIC#x^{2}-2x-4y+9=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(1,2),p=1" }, "steps": { "type": "interim", "title": "$$x^{2}-2x-4y+9=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(1,\\:2\\right),\\:$$and focal length $$|p|=1$$", "input": "x^{2}-2x-4y+9=0", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(y-k\\right)=\\left(x-h\\right)^{2}\\:$$ is the standard equation for an up-down facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$x^{2}-2x-4y+9=0\\:$$in the standard form", "input": "x^{2}-2x-4y+9=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "4y=x^{2}-2x+9" }, { "type": "step", "primary": "Divide by $$4$$", "result": "y=\\frac{x^{2}}{4}-\\frac{x}{2}+\\frac{9}{4}" }, { "type": "interim", "title": "Complete the square $$\\frac{x^{2}}{4}-\\frac{x}{2}+\\frac{9}{4}:{\\quad}\\frac{1}{4}\\left(x-1\\right)^{2}+2$$", "input": "\\frac{x^{2}}{4}-\\frac{x}{2}+\\frac{9}{4}", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c$$", "result": "\\frac{1\\cdot\\:x^{2}}{4}-\\frac{x}{2}+\\frac{9}{4}" }, { "type": "step", "primary": "Write $$\\frac{1x^{2}}{4}-\\frac{x}{2}+\\frac{9}{4}\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$\\frac{1}{4}$$" ], "result": "\\frac{1}{4}\\left(x^{2}-2x+9\\right)" }, { "type": "interim", "title": "$$2a=-2{\\quad:\\quad}a=-1$$", "input": "2a=-2", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-2", "result": "a=-1", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-2}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=-1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-1\\right)^{2}\\:$$", "result": "\\frac{1}{4}\\left(x^{2}-2x+9+\\left(-1\\right)^{2}-\\left(-1\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-2x+\\left(-1\\right)^{2}=\\left(x-1\\right)^{2}$$", "Complete the square" ], "result": "\\frac{1}{4}\\left(\\left(x-1\\right)^{2}+9-\\left(-1\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{1}{4}\\left(x-1\\right)^{2}+2" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=\\frac{1}{4}\\left(x-1\\right)^{2}+2" }, { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "y-2=\\frac{1}{4}\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$\\frac{1}{4}$$", "result": "4\\left(y-2\\right)=\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\cdot\\:1\\cdot\\:\\left(y-2\\right)=\\left(x-1\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\cdot\\:1\\cdot\\:\\left(y-2\\right)=\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(1,\\:2\\right),\\:p=1" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-1)^{2}}{4\\cdot 1}+2", "displayFormula": "4\\cdot 1(y-2)=(x-1)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=1", "displayFormula": "y=1", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,2)", "(1,3)" ], "pointsDecimal": [ { "fst": 1, "snd": 2 }, { "fst": 1, "snd": 3 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": 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