x^2+y^2-4x+10y+13=0

{ "query": { "display": "$$x^{2}+y^{2}-4x+10y+13=0$$", "symbolab_question": "CONIC#x^{2}+y^{2}-4x+10y+13=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Circle", "subTopic": "formula", "default": "(a,b)=(2,-5),r=4" }, "steps": { "type": "interim", "title": "$$x^{2}+y^{2}-4x+10y+13=0:\\quad$$Circle with center at $$\\left(2,\\:-5\\right)\\:$$and radius $$r=4$$", "input": "x^{2}+y^{2}-4x+10y+13=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 $$x^{2}+y^{2}-4x+10y+13=0\\:$$in the form of the standard circle equation", "input": "x^{2}+y^{2}-4x+10y+13=0", "steps": [ { "type": "step", "primary": "Move the loose number to the right side", "result": "x^{2}-4x+y^{2}+10y=-13" }, { "type": "step", "primary": "Group x-variables and y-variables together", "result": "\\left(x^{2}-4x\\right)+\\left(y^{2}+10y\\right)=-13" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\left(x^{2}-4x+4\\right)+\\left(y^{2}+10y\\right)=-13+4" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x-2\\right)^{2}+\\left(y^{2}+10y\\right)=-13+4" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\left(x-2\\right)^{2}+\\left(y^{2}+10y+25\\right)=-13+4+25" }, { "type": "step", "primary": "Convert to square form", "result": "\\left(x-2\\right)^{2}+\\left(y+5\\right)^{2}=-13+4+25" }, { "type": "step", "primary": "Refine $$-13+4+25$$", "result": "\\left(x-2\\right)^{2}+\\left(y+5\\right)^{2}=16" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\left(x-2\\right)^{2}+\\left(y-\\left(-5\\right)\\right)^{2}=4^{2}" } ], "meta": { "interimType": "Circle Canonical Format 1Eq" } }, { "type": "step", "result": "\\left(x-2\\right)^{2}+\\left(y-\\left(-5\\right)\\right)^{2}=4^{2}" }, { "type": "step", "primary": "Therefore the circle properties are:", "result": "\\left(a,\\:b\\right)=\\left(2,\\:-5\\right),\\:r=4" } ], "meta": { "solvingClass": "Circle" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4^{2}-(x-2)^{2}}-5", "displayFormula": "(x-2)^{2}+(y-(-5))^{2}=4^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4^{2}-(x-2)^{2}}-5", "displayFormula": "(x-2)^{2}+(y-(-5))^{2}=4^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(2,-5)" ], "pointsDecimal": [ { "fst": 2, "snd": -5 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "2", "p1y": "-5", "p2x": "4.8284271247462", "p2y": "-2.1715728752538", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "\\mathrm{radius=}4" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "(x-2)^{2}+(y-(-5))^{2}=4^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -7, "xMax": 11, "yMin": -12.8, "yMax": 5.2 } }, "showViewLarger": true } } }