9(x-1)^2-16(y+2)^2=144

{ "query": { "display": "$$9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}=144$$", "symbolab_question": "CONIC#9(x-1)^{2}-16(y+2)^{2}=144" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "formula", "default": "(h,k)=(1,-2),a=4,b=3" }, "steps": { "type": "interim", "title": "$$9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}=144:\\quad$$Right-left Hyperbola with $$\\left(h,\\:k\\right)=\\left(1,\\:-2\\right),\\:a=4,\\:b=3$$", "input": "9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}=144", "steps": [ { "type": "definition", "title": "Hyperbola standard equation", "text": "$$\\frac{\\left(x-h\\right)^{2}}{a^2}-\\frac{\\left(y-k\\right)^{2}}{b^2}=1\\:$$ is the standard equation for a right-left facing hyperbola<br/>with center $$\\bold{\\left(h,\\:k\\right)},\\:$$ semi-axis $$\\bold{a}$$ and semi-conjugate-axis $$\\bold{b}$$." }, { "type": "interim", "title": "Rewrite $$9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}=144\\:$$in the form of a standard hyperbola equation", "input": "9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}=144", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}-144=0" }, { "type": "interim", "title": "Simplify $$9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}-144:{\\quad}9x^{2}-18x-16y^{2}-64y-199$$", "input": "9\\left(x-1\\right)^{2}-16\\left(y+2\\right)^{2}-144", "result": "9x^{2}-18x-16y^{2}-64y-199=0", "steps": [ { "type": "interim", "title": "$$\\left(x-1\\right)^{2}:{\\quad}x^{2}-2x+1$$", "result": "=9\\left(x^{2}-2x+1\\right)-16\\left(y+2\\right)^{2}-144", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\cdot\\:1+1^{2}:{\\quad}x^{2}-2x+1$$", "input": "x^{2}-2x\\cdot\\:1+1^{2}", "result": "=x^{2}-2x+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=x^{2}-2\\cdot\\:1\\cdot\\:x+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=x^{2}-2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\left(y+2\\right)^{2}:{\\quad}y^{2}+4y+4$$", "result": "=9\\left(x^{2}-2x+1\\right)-16\\left(y^{2}+4y+4\\right)-144", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=y,\\:\\:b=2$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=y^{2}+2y\\cdot\\:2+2^{2}" }, { "type": "interim", "title": "Simplify $$y^{2}+2y\\cdot\\:2+2^{2}:{\\quad}y^{2}+4y+4$$", "input": "y^{2}+2y\\cdot\\:2+2^{2}", "result": "=y^{2}+4y+4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=y^{2}+4y+2^{2}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=y^{2}+4y+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$9\\left(x^{2}-2x+1\\right):{\\quad}9x^{2}-18x+9$$", "input": "9\\left(x^{2}-2x+1\\right)", "result": "=9x^{2}-18x+9-16\\left(y^{2}+4y+4\\right)-144", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=9x^{2}+9\\left(-2x\\right)+9\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=9x^{2}-9\\cdot\\:2x+9\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$9x^{2}-9\\cdot\\:2x+9\\cdot\\:1:{\\quad}9x^{2}-18x+9$$", "input": "9x^{2}-9\\cdot\\:2x+9\\cdot\\:1", "result": "=9x^{2}-18x+9", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:2=18$$", "result": "=9x^{2}-18x+9\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:1=9$$", "result": "=9x^{2}-18x+9" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7a2+qIh9GFuK0gFz++foWNi061ljBSPJeENOw2efoSWv24nUJxkcX8qJh45tDIA/vOywRgDR2u8feEqvsXE3aqnql8XXPq6bNQlMm+36iNhnSmTajmUvFzkMoCzW0lT2ekNSZegtLGHdeCBAEWmwE/g==" } }, { "type": "interim", "title": "Expand $$-16\\left(y^{2}+4y+4\\right):{\\quad}-16y^{2}-64y-64$$", "input": "-16\\left(y^{2}+4y+4\\right)", "result": "=9x^{2}-18x+9-16y^{2}-64y-64-144", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=\\left(-16\\right)y^{2}+\\left(-16\\right)\\cdot\\:4y+\\left(-16\\right)\\cdot\\:4", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-16y^{2}-16\\cdot\\:4y-16\\cdot\\:4" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:4=64$$", "result": "=-16y^{2}-64y-64" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OQHtlvvoSjarHP+drs6KL9MvHyY50dhXPFfrjcmooUifsDYkpM709GEZ3Q8wzj2NPuq4K0alRmIN5FGuHvfMT0UqTd96MWTKI6Kr2Ib0iQBM8N9KbDwRIspDTl7qRiXTL50PqXMFmUwf16ARoajewA==" } }, { "type": "interim", "title": "Simplify $$9x^{2}-18x+9-16y^{2}-64y-64-144:{\\quad}9x^{2}-18x-16y^{2}-64y-199$$", "input": "9x^{2}-18x+9-16y^{2}-64y-64-144", "result": "=9x^{2}-18x-16y^{2}-64y-199", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=9x^{2}-18x-16y^{2}-64y+9-64-144" }, { "type": "step", "primary": "Add/Subtract the numbers: $$9-64-144=-199$$", "result": "=9x^{2}-18x-16y^{2}-64y-199" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Add $$199$$ to both sides", "result": "9x^{2}-18x-16y^{2}-64y=199" }, { "type": "step", "primary": "Factor out coefficient of square terms", "result": "9\\left(x^{2}-2x\\right)-16\\left(y^{2}+4y\\right)=199" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$9$$", "result": "\\left(x^{2}-2x\\right)-\\frac{16}{9}\\left(y^{2}+4y\\right)=\\frac{199}{9}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$16$$", "result": "\\frac{1}{16}\\left(x^{2}-2x\\right)-\\frac{1}{9}\\left(y^{2}+4y\\right)=\\frac{199}{144}" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "\\frac{1}{16}\\left(x^{2}-2x+1\\right)-\\frac{1}{9}\\left(y^{2}+4y\\right)=\\frac{199}{144}+\\frac{1}{16}\\left(1\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "\\frac{1}{16}\\left(x-1\\right)^{2}-\\frac{1}{9}\\left(y^{2}+4y\\right)=\\frac{199}{144}+\\frac{1}{16}\\left(1\\right)" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "\\frac{1}{16}\\left(x-1\\right)^{2}-\\frac{1}{9}\\left(y^{2}+4y+4\\right)=\\frac{199}{144}+\\frac{1}{16}\\left(1\\right)-\\frac{1}{9}\\left(4\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "\\frac{1}{16}\\left(x-1\\right)^{2}-\\frac{1}{9}\\left(y+2\\right)^{2}=\\frac{199}{144}+\\frac{1}{16}\\left(1\\right)-\\frac{1}{9}\\left(4\\right)" }, { "type": "step", "primary": "Refine $$\\frac{199}{144}+\\frac{1}{16}\\left(1\\right)-\\frac{1}{9}\\left(4\\right)$$", "result": "\\frac{1}{16}\\left(x-1\\right)^{2}-\\frac{1}{9}\\left(y+2\\right)^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{\\left(x-1\\right)^{2}}{16}-\\frac{\\left(y+2\\right)^{2}}{9}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-1\\right)^{2}}{4^{2}}-\\frac{\\left(y-\\left(-2\\right)\\right)^{2}}{3^{2}}=1" } ], "meta": { "interimType": "Hyperbola Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(x-1\\right)^{2}}{4^{2}}-\\frac{\\left(y-\\left(-2\\right)\\right)^{2}}{3^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(1,\\:-2\\right),\\:a=4,\\:b=3" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{3}{4}(x-1)-2", "displayFormula": "y=\\frac{3}{4}(x-1)-2", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{3}{4}(x-1)-2", "displayFormula": "y=-\\frac{3}{4}(x-1)-2", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{9(\\frac{(x-1)^{2}}{4^{2}}-1)}-2", "displayFormula": "\\frac{(x-1)^{2}}{4^{2}}-\\frac{(y-(-2))^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{9(\\frac{(x-1)^{2}}{4^{2}}-1)}-2", "displayFormula": "\\frac{(x-1)^{2}}{4^{2}}-\\frac{(y-(-2))^{2}}{3^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,-2)" ], "pointsDecimal": [ { "fst": 1, "snd": -2 } ], 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