x^2-2x+y^2+2y=7

{ "query": { "display": "$$x^{2}-2x+y^{2}+2y=7$$", "symbolab_question": "CONIC#x^{2}-2x+y^{2}+2y=7" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Circle", "subTopic": "formula", "default": "(a,b)=(1,-1),r=3" }, "steps": { "type": "interim", "title": "$$x^{2}-2x+y^{2}+2y=7:\\quad$$Circle with center at $$\\left(1,\\:-1\\right)\\:$$and radius $$r=3$$", "input": "x^{2}-2x+y^{2}+2y=7", "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 $$x^{2}-2x+y^{2}+2y=7\\:$$in the form of the standard circle equation", "input": "x^{2}-2x+y^{2}+2y=7", "steps": [ { "type": "step", "primary": "Group x-variables and y-variables together", "result": "\\left(x^{2}-2x\\right)+\\left(y^{2}+2y\\right)=7" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\left(x^{2}-2x+1\\right)+\\left(y^{2}+2y\\right)=7+1" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x-1\\right)^{2}+\\left(y^{2}+2y\\right)=7+1" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\left(x-1\\right)^{2}+\\left(y^{2}+2y+1\\right)=7+1+1" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x-1\\right)^{2}+\\left(y+1\\right)^{2}=7+1+1" }, { "type": "step", "primary": "Refine $$7+1+1$$", "result": "\\left(x-1\\right)^{2}+\\left(y+1\\right)^{2}=9" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\left(x-1\\right)^{2}+\\left(y-\\left(-1\\right)\\right)^{2}=3^{2}" } ], "meta": { "interimType": "Circle Canonical Format 1Eq" } }, { "type": "step", "result": "\\left(x-1\\right)^{2}+\\left(y-\\left(-1\\right)\\right)^{2}=3^{2}" }, { "type": "step", "primary": "Therefore the circle properties are:", "result": "\\left(a,\\:b\\right)=\\left(1,\\:-1\\right),\\:r=3" } ], "meta": { "solvingClass": "Circle" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{3^{2}-(x-1)^{2}}-1", "displayFormula": "(x-1)^{2}+(y-(-1))^{2}=3^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{3^{2}-(x-1)^{2}}-1", "displayFormula": "(x-1)^{2}+(y-(-1))^{2}=3^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,-1)" ], "pointsDecimal": [ { "fst": 1, "snd": -1 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "1", "p1y": "-1", "p2x": "3.1213203435596", "p2y": "1.1213203435596", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "\\mathrm{radius=}3" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "(x-1)^{2}+(y-(-1))^{2}=3^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -5.75, "xMax": 7.75, "yMin": -7.75, "yMax": 5.75 } }, "showViewLarger": true } } }