directrix y^2+12x+4y-32=0

{ "query": { "display": "directrix $$y^{2}+12x+4y-32=0$$", "symbolab_question": "CONIC#directrix y^{2}+12x+4y-32=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "directrix", "default": "x=6" }, "steps": { "type": "interim", "title": "Parabola directrix given $$y^{2}+12x+4y-32=0:{\\quad}x=6$$", "steps": [ { "type": "definition", "title": "Parabola Directrix", "text": "A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (the directrix)" }, { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(x-h\\right)=\\left(y-k\\right)^{2}\\:$$ is the standard equation for a right-left facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$y^{2}+12x+4y-32=0\\:$$in the standard form", "input": "y^{2}+12x+4y-32=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "12x=-y^{2}-4y+32" }, { "type": "step", "primary": "Divide by $$12$$", "result": "x=-\\frac{y^{2}}{12}-\\frac{y}{3}+\\frac{8}{3}" }, { "type": "interim", "title": "Complete the square $$-\\frac{y^{2}}{12}-\\frac{y}{3}+\\frac{8}{3}:{\\quad}-\\frac{1}{12}\\left(y+2\\right)^{2}+3$$", "input": "-\\frac{y^{2}}{12}-\\frac{y}{3}+\\frac{8}{3}", "steps": [ { "type": "step", "primary": "Write $$-\\frac{y^{2}}{12}-\\frac{y}{3}+\\frac{8}{3}\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-\\frac{1}{12}$$" ], "result": "-\\frac{1}{12}\\left(y^{2}+4y-32\\right)" }, { "type": "interim", "title": "$$2a=4{\\quad:\\quad}a=2$$", "input": "2a=4", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=4", "result": "a=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{4}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=2" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$2^{2}\\:$$", "result": "-\\frac{1}{12}\\left(y^{2}+4y-32+2^{2}-2^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$y^{2}+4y+2^{2}=\\left(y+2\\right)^{2}$$", "Complete the square" ], "result": "-\\frac{1}{12}\\left(\\left(y+2\\right)^{2}-32-2^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{1}{12}\\left(y+2\\right)^{2}+3" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "x=-\\frac{1}{12}\\left(y+2\\right)^{2}+3" }, { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "x-3=-\\frac{1}{12}\\left(y+2\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-\\frac{1}{12}$$", "result": "-12\\left(x-3\\right)=\\left(y+2\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-3\\right)\\left(x-3\\right)=\\left(y-\\left(-2\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-3\\right)\\left(x-3\\right)=\\left(y-\\left(-2\\right)\\right)^{2}" }, { "type": "step", "result": "\\left(h,\\:k\\right)=\\left(3,\\:-2\\right),\\:p=-3" }, { "type": "step", "primary": "Parabola is symmetric around the x-axis and so the directrix is a line parallel to the y-axis, a distance $$-p$$ from the center $$\\left(3,\\:-2\\right)$$ x-coordinate ", "result": "x=3-p" }, { "type": "step", "result": "x=3-\\left(-3\\right)" }, { "type": "step", "primary": "Refine", "result": "x=6" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-3)(x-3)}-2", "displayFormula": "4(-3)(x-3)=(y-(-2))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-3)(x-3)}-2", "displayFormula": "4(-3)(x-3)=(y-(-2))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=6", "displayFormula": "x=6", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ 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