vertices 9x^2+4y^2+36x-24y+36=0

{ "query": { "display": "vertices $$9x^{2}+4y^{2}+36x-24y+36=0$$", "symbolab_question": "CONIC#vertices 9x^{2}+4y^{2}+36x-24y+36=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Ellipse", "subTopic": "vertices", "default": "(-2,6),(-2,0)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Ellipse vertices given $$9x^{2}+4y^{2}+36x-24y+36=0:{\\quad}\\left(-2,\\:6\\right),\\:\\left(-2,\\:0\\right)$$", "steps": [ { "type": "definition", "title": "Ellipse vertices", "text": "The vertices are the two points on the ellipse that intersect the major axis<br/>For an ellipse with major axis parallel to the y-axis, the vertices are $$\\left(h,\\:k+b\\right),\\:\\left(h,\\:k-b\\right)$$" }, { "type": "step", "result": "\\left(h,\\:k+b\\right),\\:\\left(h,\\:k-b\\right)" }, { "type": "step", "primary": "Calculate ellipse properties" }, { "type": "interim", "title": "$$9x^{2}+4y^{2}+36x-24y+36=0:{\\quad}$$Ellipse with center $$\\left(h,\\:k\\right)=\\left(-2,\\:3\\right),\\:b=3,\\:a=2$$", "input": "9x^{2}+4y^{2}+36x-24y+36=0", "steps": [ { "type": "definition", "title": "Ellipse standard equation", "text": "$$\\frac{\\left(x-h\\right)^{2}}{a^2}+\\frac{\\left(y-k\\right)^{2}}{b^2}=1\\:$$is the ellipse standard equation<br/>with center $$\\left(h,\\:k\\right)\\:$$and $$a,\\:b$$ are the semi-major and semi-minor axes" }, { "type": "interim", "title": "Rewrite $$9x^{2}+4y^{2}+36x-24y+36=0\\:$$in the form of the standard ellipse equation", "input": "9x^{2}+4y^{2}+36x-24y+36=0", "steps": [ { "type": "step", "primary": "Subtract $$36$$ from both sides", "result": "9x^{2}+36x+4y^{2}-24y=-36" }, { "type": "step", "primary": "Factor out coefficient of square terms", "result": "9\\left(x^{2}+4x\\right)+4\\left(y^{2}-6y\\right)=-36" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$9$$", "result": "\\left(x^{2}+4x\\right)+\\frac{4}{9}\\left(y^{2}-6y\\right)=-4" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$4$$", "result": "\\frac{1}{4}\\left(x^{2}+4x\\right)+\\frac{1}{9}\\left(y^{2}-6y\\right)=-1" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\frac{1}{4}\\left(x^{2}+4x+4\\right)+\\frac{1}{9}\\left(y^{2}-6y\\right)=-1+\\frac{1}{4}\\left(4\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "\\frac{1}{4}\\left(x+2\\right)^{2}+\\frac{1}{9}\\left(y^{2}-6y\\right)=-1+\\frac{1}{4}\\left(4\\right)" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\frac{1}{4}\\left(x+2\\right)^{2}+\\frac{1}{9}\\left(y^{2}-6y+9\\right)=-1+\\frac{1}{4}\\left(4\\right)+\\frac{1}{9}\\left(9\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "\\frac{1}{4}\\left(x+2\\right)^{2}+\\frac{1}{9}\\left(y-3\\right)^{2}=-1+\\frac{1}{4}\\left(4\\right)+\\frac{1}{9}\\left(9\\right)" }, { "type": "step", "primary": "Refine $$-1+\\frac{1}{4}\\left(4\\right)+\\frac{1}{9}\\left(9\\right)$$", "result": "\\frac{1}{4}\\left(x+2\\right)^{2}+\\frac{1}{9}\\left(y-3\\right)^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{\\left(x+2\\right)^{2}}{4}+\\frac{\\left(y-3\\right)^{2}}{9}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-\\left(-2\\right)\\right)^{2}}{2^{2}}+\\frac{\\left(y-3\\right)^{2}}{3^{2}}=1" } ], "meta": { "interimType": "Ellipse Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(x-\\left(-2\\right)\\right)^{2}}{2^{2}}+\\frac{\\left(y-3\\right)^{2}}{3^{2}}=1" }, { "type": "step", "primary": "Therefore ellipse properties are:", "result": "\\left(h,\\:k\\right)=\\left(-2,\\:3\\right),\\:a=2,\\:b=3" }, { "type": "step", "primary": "$$b>a\\:$$therefore $$b\\:$$is semi-major axis and $$a\\:$$is semi-minor axis", "result": "\\mathrm{Ellipse\\:with\\:center}\\:\\left(h,\\:k\\right)=\\left(-2,\\:3\\right),\\:b=3,\\:a=2" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "\\left(-2,\\:3+3\\right),\\:\\left(-2,\\:3-3\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(-2,\\:6\\right),\\:\\left(-2,\\:0\\right)" } ], "meta": { "solvingClass": "Ellipse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{9(1-\\frac{(x-(-2))^{2}}{2^{2}})}+3", "displayFormula": "\\frac{(x-(-2))^{2}}{2^{2}}+\\frac{(y-3)^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{9(1-\\frac{(x-(-2))^{2}}{2^{2}})}+3", "displayFormula": "\\frac{(x-(-2))^{2}}{2^{2}}+\\frac{(y-3)^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-2,3)", "(-2,6)", "(-2,0)" ], "pointsDecimal": [ { "fst": -2, "snd": 3 }, { "fst": -2, "snd": 6 }, { "fst": -2, "snd": 0 } ], "attributes": [ { "color": 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