5x+3y^2=0

{ "query": { "display": "$$5x+3y^{2}=0$$", "symbolab_question": "CONIC#5x+3y^{2}=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,0),p=-\\frac{5}{12}" }, "steps": { "type": "interim", "title": "$$5x+3y^{2}=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:$$and focal length $$|p|=\\frac{5}{12}$$", "input": "5x+3y^{2}=0", "steps": [ { "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 $$5x+3y^{2}=0\\:$$in the standard form:$${\\quad}4\\left(-\\frac{5}{12}\\right)\\left(x-0\\right)=\\left(y-0\\right)^{2}$$", "input": "5x+3y^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$3y^{2}$$ from both sides", "result": "5x+3y^{2}-3y^{2}=0-3y^{2}" }, { "type": "step", "primary": "Refine", "result": "5x=-3y^{2}" }, { "type": "step", "primary": "Divide both sides by $$-3$$", "result": "\\frac{5x}{-3}=\\frac{-3y^{2}}{-3}" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{5x}{3}=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-\\frac{5}{3}}{4}x=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-\\frac{5}{12}\\right)x=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-\\frac{5}{12}\\right)\\left(x-0\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kXWGCo61GcOAJ7H6AkpWd9g4YQc/BH29hElCH28KEycjGsBIMn028UBVvff51AwIUo+0skB3JhOoVsRyoC4GXjh2no6HYikYL/uXZaE/9Y+RUZjadPlpTubkRAByJR5EWGSaISCMW1S3mZs3994Dl4dEeJB8NSwK2cnf2Bc1WE3bdeLci9PfzmQBgUrYJA7E6UOYgVBOOp0yPwtCb5qnM3QUgoS87LFk0WHEX4+Itv8W/3uc7VtdmRaH03Lx4dDusIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-\\frac{5}{12}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-\\frac{5}{12})x}+0", "displayFormula": "4(-\\frac{5}{12})x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-\\frac{5}{12})x}+0", "displayFormula": "4(-\\frac{5}{12})x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{5}{12}", "displayFormula": "x=\\frac{5}{12}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(-\\frac{5}{12},0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": -0.4166666666666667, "snd": 0 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-\\frac{5}{12})(x)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -4.6875, "xMax": 4.6875, "yMin": -4.6875, "yMax": 4.6875 } }, "showViewLarger": true } } }