x^2+y^2+6x-4y-12=0

{ "query": { "display": "$$x^{2}+y^{2}+6x-4y-12=0$$", "symbolab_question": "CONIC#x^{2}+y^{2}+6x-4y-12=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Circle", "subTopic": "formula", "default": "(a,b)=(-3,2),r=5" }, "steps": { "type": "interim", "title": "$$x^{2}+y^{2}+6x-4y-12=0:\\quad$$Circle with center at $$\\left(-3,\\:2\\right)\\:$$and radius $$r=5$$", "input": "x^{2}+y^{2}+6x-4y-12=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 $$x^{2}+y^{2}+6x-4y-12=0\\:$$in the form of the standard circle equation", "input": "x^{2}+y^{2}+6x-4y-12=0", "steps": [ { "type": "step", "primary": "Move the loose number to the right side", "result": "x^{2}+6x+y^{2}-4y=12" }, { "type": "step", "primary": "Group x-variables and y-variables together", "result": "\\left(x^{2}+6x\\right)+\\left(y^{2}-4y\\right)=12" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\left(x^{2}+6x+9\\right)+\\left(y^{2}-4y\\right)=12+9" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x+3\\right)^{2}+\\left(y^{2}-4y\\right)=12+9" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\left(x+3\\right)^{2}+\\left(y^{2}-4y+4\\right)=12+9+4" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x+3\\right)^{2}+\\left(y-2\\right)^{2}=12+9+4" }, { "type": "step", "primary": "Refine $$12+9+4$$", "result": "\\left(x+3\\right)^{2}+\\left(y-2\\right)^{2}=25" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\left(x-\\left(-3\\right)\\right)^{2}+\\left(y-2\\right)^{2}=5^{2}" } ], "meta": { "interimType": "Circle Canonical Format 1Eq" } }, { "type": "step", "result": "\\left(x-\\left(-3\\right)\\right)^{2}+\\left(y-2\\right)^{2}=5^{2}" }, { "type": "step", "primary": "Therefore the circle properties are:", "result": "\\left(a,\\:b\\right)=\\left(-3,\\:2\\right),\\:r=5" } ], "meta": { "solvingClass": "Circle" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{5^{2}-(x+3)^{2}}+2", "displayFormula": "(x-(-3))^{2}+(y-2)^{2}=5^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{5^{2}-(x+3)^{2}}+2", "displayFormula": "(x-(-3))^{2}+(y-2)^{2}=5^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-3,2)" ], "pointsDecimal": [ { "fst": -3, "snd": 2 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "-3", "p1y": "2", "p2x": "0.5355339059327", "p2y": "5.5355339059327", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "\\mathrm{radius=}5" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "(x-(-3))^{2}+(y-2)^{2}=5^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -14.25, "xMax": 8.25, "yMin": -9.25, "yMax": 13.25 } }, "showViewLarger": true } } }