x^2+y^2+2x+6y+2=0

{ "query": { "display": "$$x^{2}+y^{2}+2x+6y+2=0$$", "symbolab_question": "CONIC#x^{2}+y^{2}+2x+6y+2=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Circle", "subTopic": "formula", "default": "(a,b)=(-1,-3),r=2\\sqrt{2}" }, "steps": { "type": "interim", "title": "$$x^{2}+y^{2}+2x+6y+2=0:\\quad$$Circle with center at $$\\left(-1,\\:-3\\right)\\:$$and radius $$r=2\\sqrt{2}$$", "input": "x^{2}+y^{2}+2x+6y+2=0", "steps": [ { "type": "definition", "title": "Circle Equation", "text": "$$\\left(x−a\\right)^2+\\left(y−b\\right)^2=r^2\\:\\:$$is the circle equation with a radius r, centered at $$\\left(a,\\:b\\right)$$" }, { "type": "interim", "title": "Rewrite $$x^{2}+y^{2}+2x+6y+2=0\\:$$in the form of the standard circle equation", "input": "x^{2}+y^{2}+2x+6y+2=0", "steps": [ { "type": "step", "primary": "Move the loose number to the right side", "result": "x^{2}+2x+y^{2}+6y=-2" }, { "type": "step", "primary": "Group x-variables and y-variables together", "result": "\\left(x^{2}+2x\\right)+\\left(y^{2}+6y\\right)=-2" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\left(x^{2}+2x+1\\right)+\\left(y^{2}+6y\\right)=-2+1" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x+1\\right)^{2}+\\left(y^{2}+6y\\right)=-2+1" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\left(x+1\\right)^{2}+\\left(y^{2}+6y+9\\right)=-2+1+9" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x+1\\right)^{2}+\\left(y+3\\right)^{2}=-2+1+9" }, { "type": "step", "primary": "Refine $$-2+1+9$$", "result": "\\left(x+1\\right)^{2}+\\left(y+3\\right)^{2}=8" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\left(x-\\left(-1\\right)\\right)^{2}+\\left(y-\\left(-3\\right)\\right)^{2}=\\left(2\\sqrt{2}\\right)^{2}" } ], "meta": { "interimType": "Circle Canonical Format 1Eq" } }, { "type": "step", "result": "\\left(x-\\left(-1\\right)\\right)^{2}+\\left(y-\\left(-3\\right)\\right)^{2}=\\left(2\\sqrt{2}\\right)^{2}" }, { "type": "step", "primary": "Therefore the circle properties are:", "result": "\\left(a,\\:b\\right)=\\left(-1,\\:-3\\right),\\:r=2\\sqrt{2}" } ], "meta": { "solvingClass": "Circle" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{(2\\sqrt{2})^{2}-(x+1)^{2}}-3", "displayFormula": "(x-(-1))^{2}+(y-(-3))^{2}=(2\\sqrt{2})^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{(2\\sqrt{2})^{2}-(x+1)^{2}}-3", "displayFormula": "(x-(-1))^{2}+(y-(-3))^{2}=(2\\sqrt{2})^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-1,-3)" ], "pointsDecimal": [ { "fst": -1, "snd": -3 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "-1", "p1y": "-3", "p2x": "1", "p2y": "-1", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "\\mathrm{radius=}2\\sqrt{2}" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "(x-(-1))^{2}+(y-(-3))^{2}=(2\\sqrt{2})^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -7.3639610306789285, "xMax": 5.3639610306789285, "yMin": -8.821452268073777, "yMax": 3.9064697932840775 } }, "showViewLarger": true } } }