-x^2+12x-12y-96=0

{ "query": { "display": "$$-x^{2}+12x-12y-96=0$$", "symbolab_question": "CONIC#-x^{2}+12x-12y-96=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(6,-5),p=-3" }, "steps": { "type": "interim", "title": "$$-x^{2}+12x-12y-96=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(6,\\:-5\\right),\\:$$and focal length $$|p|=3$$", "input": "-x^{2}+12x-12y-96=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 $$-x^{2}+12x-12y-96=0\\:$$in the standard form", "input": "-x^{2}+12x-12y-96=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "12y=-x^{2}+12x-96" }, { "type": "step", "primary": "Divide by $$12$$", "result": "y=-\\frac{x^{2}}{12}+x-8" }, { "type": "interim", "title": "Complete the square $$-\\frac{x^{2}}{12}+x-8:{\\quad}-\\frac{1}{12}\\left(x-6\\right)^{2}-5$$", "input": "-\\frac{x^{2}}{12}+x-8", "steps": [ { "type": "step", "primary": "Write $$-\\frac{x^{2}}{12}+x-8\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-\\frac{1}{12}$$" ], "result": "-\\frac{1}{12}\\left(x^{2}-12x+96\\right)" }, { "type": "interim", "title": "$$2a=-12{\\quad:\\quad}a=-6$$", "input": "2a=-12", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-12", "result": "a=-6", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-12}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2a}{2}=\\frac{-12}{2}", "result": "a=-6", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2a}{2}:{\\quad}a$$", "input": "\\frac{2a}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=a" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n/9TIosDfOyZXH4usT9jsy061ljBSPJeENOw2efoSWuFFDuzzgIzA1F36c+SAFGgRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pQyGeXuoJHAaVrX92ZjOTT" } }, { "type": "interim", "title": "Simplify $$\\frac{-12}{2}:{\\quad}-6$$", "input": "\\frac{-12}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{12}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{2}=6$$", "result": "=-6" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GKWtQW4fbFStcswMIqzf9QCWKUbvV6WK3fDUgFtg3Q8IVlRpyxl4O+CcU61LkQcQZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0s7e1hziQhbs6o+Enn0k2JsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "a=-6" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-6\\right)^{2}\\:$$", "result": "-\\frac{1}{12}\\left(x^{2}-12x+96+\\left(-6\\right)^{2}-\\left(-6\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-12x+\\left(-6\\right)^{2}=\\left(x-6\\right)^{2}$$", "Complete the square" ], "result": "-\\frac{1}{12}\\left(\\left(x-6\\right)^{2}+96-\\left(-6\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{1}{12}\\left(x-6\\right)^{2}-5" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=-\\frac{1}{12}\\left(x-6\\right)^{2}-5" }, { "type": "step", "primary": "Add $$5$$ to both sides", "result": "y+5=-\\frac{1}{12}\\left(x-6\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-\\frac{1}{12}$$", "result": "-12\\left(y+5\\right)=\\left(x-6\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-3\\right)\\left(y-\\left(-5\\right)\\right)=\\left(x-6\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-3\\right)\\left(y-\\left(-5\\right)\\right)=\\left(x-6\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(6,\\:-5\\right),\\:p=-3" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-6)^{2}}{4(-3)}-5", "displayFormula": "4(-3)(y-(-5))=(x-6)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-2", "displayFormula": "y=-2", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(6,-5)", "(6,-8)" ], "pointsDecimal": [ { "fst": 6, "snd": -5 }, { "fst": 6, "snd": -8 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ 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