radius x^2+y^2-18x-14y+124=0

{ "query": { "display": "radius $$x^{2}+y^{2}-18x-14y+124=0$$", "symbolab_question": "CONIC#radius x^{2}+y^{2}-18x-14y+124=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Circle", "subTopic": "radius", "default": "r=\\sqrt{6}" }, "steps": { "type": "interim", "title": "Radius given $$x^{2}+y^{2}-18x-14y+124=0:{\\quad}r=\\sqrt{6}$$", "input": "x^{2}+y^{2}-18x-14y+124=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 $$x^{2}+y^{2}-18x-14y+124=0\\:$$in the form of the standard circle equation", "input": "x^{2}+y^{2}-18x-14y+124=0", "steps": [ { "type": "step", "primary": "Move the loose number to the right side", "result": "x^{2}-18x+y^{2}-14y=-124" }, { "type": "step", "primary": "Group x-variables and y-variables together", "result": "\\left(x^{2}-18x\\right)+\\left(y^{2}-14y\\right)=-124" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\left(x^{2}-18x+81\\right)+\\left(y^{2}-14y\\right)=-124+81" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x-9\\right)^{2}+\\left(y^{2}-14y\\right)=-124+81" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\left(x-9\\right)^{2}+\\left(y^{2}-14y+49\\right)=-124+81+49" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x-9\\right)^{2}+\\left(y-7\\right)^{2}=-124+81+49" }, { "type": "step", "primary": "Refine $$-124+81+49$$", "result": "\\left(x-9\\right)^{2}+\\left(y-7\\right)^{2}=6" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\left(x-9\\right)^{2}+\\left(y-7\\right)^{2}=\\left(\\sqrt{6}\\right)^{2}" } ], "meta": { "interimType": "Circle Canonical Format 1Eq" } }, { "type": "step", "result": "\\left(x-9\\right)^{2}+\\left(y-7\\right)^{2}=\\left(\\sqrt{6}\\right)^{2}" }, { "type": "step", "primary": "Therefore the circle properties are:", "result": "\\left(a,\\:b\\right)=\\left(9,\\:7\\right),\\:r=\\sqrt{6}" }, { "type": "step", "primary": "And the radius is:", "result": "r=\\sqrt{6}" } ], "meta": { "solvingClass": "Circle" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{(\\sqrt{6})^{2}-(x-9)^{2}}+7", "displayFormula": "(x-9)^{2}+(y-7)^{2}=\\sqrt{6}^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{(\\sqrt{6})^{2}-(x-9)^{2}}+7", "displayFormula": "(x-9)^{2}+(y-7)^{2}=\\sqrt{6}^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(9,7)" ], "pointsDecimal": [ { "fst": 9, "snd": 7 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "9", "p1y": "7", "p2x": "10.7320508075689", "p2y": "8.7320508075689", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "\\mathrm{radius=}\\sqrt{6}" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "(x-9)^{2}+(y-7)^{2}=(\\sqrt{6})^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -1.6356413918261676, "xMax": 11.449489742783179, "yMin": -2.4927842489690253, "yMax": 10.592346885640321 } }, "showViewLarger": true } } }