((y+1)^2)/4-(x-5)^2=4

{ "query": { "display": "$$\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}=4$$", "symbolab_question": "CONIC#\\frac{(y+1)^{2}}{4}-(x-5)^{2}=4" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "formula", "default": "(h,k)=(5,-1),a=4,b=2" }, "steps": { "type": "interim", "title": "$$\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}=4:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(5,\\:-1\\right),\\:a=4,\\:b=2$$", "input": "\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}=4", "steps": [ { "type": "definition", "title": "Hyperbola standard equation", "text": "$$\\frac{\\left(y-k\\right)^{2}}{a^2}-\\frac{\\left(x-h\\right)^{2}}{b^2}=1\\:$$ is the standard equation for an up-down facing hyperbola<br/>with center $$\\bold{\\left(h,\\:k\\right)},\\:$$ semi-axis $$\\bold{a}$$ and semi-conjugate-axis $$\\bold{b}$$." }, { "type": "interim", "title": "Rewrite $$\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}=4\\:$$in the form of a standard hyperbola equation", "input": "\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}=4", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}-4=0" }, { "type": "interim", "title": "Simplify $$\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}-4:{\\quad}\\frac{-4x^{2}+40x+y^{2}+2y-115}{4}$$", "input": "\\frac{\\left(y+1\\right)^{2}}{4}-\\left(x-5\\right)^{2}-4", "result": "\\frac{-4x^{2}+40x+y^{2}+2y-115}{4}=0", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\left(x-5\\right)^{2}=\\frac{\\left(x-5\\right)^{2}4}{4},\\:4=\\frac{4\\cdot\\:4}{4}$$", "result": "=\\frac{\\left(y+1\\right)^{2}}{4}-\\frac{\\left(x-5\\right)^{2}\\cdot\\:4}{4}-\\frac{4\\cdot\\:4}{4}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\left(y+1\\right)^{2}-\\left(x-5\\right)^{2}\\cdot\\:4-4\\cdot\\:4}{4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=\\frac{\\left(y+1\\right)^{2}-4\\left(x-5\\right)^{2}-16}{4}" }, { "type": "interim", "title": "Expand $$\\left(y+1\\right)^{2}-\\left(x-5\\right)^{2}\\cdot\\:4-16:{\\quad}-4x^{2}+40x+y^{2}+2y-115$$", "input": "\\left(y+1\\right)^{2}-\\left(x-5\\right)^{2}\\cdot\\:4-16", "result": "=\\frac{-4x^{2}+40x+y^{2}+2y-115}{4}", "steps": [ { "type": "step", "result": "=\\left(y+1\\right)^{2}-4\\left(x-5\\right)^{2}-16" }, { "type": "interim", "title": "$$\\left(y+1\\right)^{2}:{\\quad}y^{2}+2y+1$$", "result": "=y^{2}+2y+1-\\left(x-5\\right)^{2}\\cdot\\:4-16", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=y,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=y^{2}+2y\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$y^{2}+2y\\cdot\\:1+1^{2}:{\\quad}y^{2}+2y+1$$", "input": "y^{2}+2y\\cdot\\:1+1^{2}", "result": "=y^{2}+2y+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=y^{2}+2\\cdot\\:1\\cdot\\:y+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=y^{2}+2y+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\left(x-5\\right)^{2}:{\\quad}x^{2}-10x+25$$", "result": "=y^{2}+2y+1-\\left(x^{2}-10x+25\\right)\\cdot\\:4-16", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=5$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\cdot\\:5+5^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\cdot\\:5+5^{2}:{\\quad}x^{2}-10x+25$$", "input": "x^{2}-2x\\cdot\\:5+5^{2}", "result": "=x^{2}-10x+25", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=x^{2}-10x+5^{2}" }, { "type": "step", "primary": "$$5^{2}=25$$", "result": "=x^{2}-10x+25" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$-4\\left(x^{2}-10x+25\\right):{\\quad}-4x^{2}+40x-100$$", "input": "-4\\left(x^{2}-10x+25\\right)", "result": "=y^{2}+2y+1-4x^{2}+40x-100-16", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=\\left(-4\\right)x^{2}+\\left(-4\\right)\\left(-10x\\right)+\\left(-4\\right)\\cdot\\:25", "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": [ "$$+\\left(-a\\right)=-a,\\:\\:\\left(-a\\right)\\left(-b\\right)=ab$$" ], "result": "=-4x^{2}+4\\cdot\\:10x-4\\cdot\\:25" }, { "type": "interim", "title": "Simplify $$-4x^{2}+4\\cdot\\:10x-4\\cdot\\:25:{\\quad}-4x^{2}+40x-100$$", "input": "-4x^{2}+4\\cdot\\:10x-4\\cdot\\:25", "result": "=-4x^{2}+40x-100", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:10=40$$", "result": "=-4x^{2}+40x-4\\cdot\\:25" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:25=100$$", "result": "=-4x^{2}+40x-100" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7byxQNagdks54ClvoZX80uqVnANiuhwjse08ef+lqfd15tMpJTBBccUWkSyvMe1SpO0AubU5mrQX7Bj4Dht7P1qN6Hv6MoTMtvtU0IQwXdn/uHDG2fzooZkB281z8T5Y+MlRWjq0tBO56Ut15zElU2r8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "Simplify $$y^{2}+2y+1-4x^{2}+40x-100-16:{\\quad}-4x^{2}+40x+y^{2}+2y-115$$", "input": "y^{2}+2y+1-4x^{2}+40x-100-16", "result": "=-4x^{2}+40x+y^{2}+2y-115", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-4x^{2}+40x+y^{2}+2y+1-100-16" }, { "type": "step", "primary": "Add/Subtract the numbers: $$1-100-16=-115$$", "result": "=-4x^{2}+40x+y^{2}+2y-115" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MVPB2n3zvzEq2vcOosJ04PalIHReApgPHLOLUt1mKAwAlilG71elit3w1IBbYN0PPHRxC30xT6jxQFN6m9ssvlMCXjdz0yMVe0anoSid6v0iAvXPV7/u/zRE4gCDqAu3EnJ/zNrC2ARoMQpaswsTNgOH/WDZtu6CtaHAngzXNoX+5sNbtNuvN0ML+EYBv632gFTeWNeBNsXUNio77Ls+nA==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Multiply by $$4$$", "result": "-4x^{2}+40x+y^{2}+2y-115=0" }, { "type": "step", "primary": "Add $$115$$ to both sides", "result": "-4x^{2}+40x+y^{2}+2y=115" }, { "type": "step", "primary": "Factor out coefficient of square terms", "result": "-4\\left(x^{2}-10x\\right)+\\left(y^{2}+2y\\right)=115" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$4$$", "result": "-\\left(x^{2}-10x\\right)+\\frac{1}{4}\\left(y^{2}+2y\\right)=\\frac{115}{4}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$1$$", "result": "-\\frac{1}{1}\\left(x^{2}-10x\\right)+\\frac{1}{4}\\left(y^{2}+2y\\right)=\\frac{115}{4}" }, { "type": "step", "primary": "Convert $$x\\:$$to square form", "result": "-\\frac{1}{1}\\left(x^{2}-10x+25\\right)+\\frac{1}{4}\\left(y^{2}+2y\\right)=\\frac{115}{4}-\\frac{1}{1}\\left(25\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "-\\frac{1}{1}\\left(x-5\\right)^{2}+\\frac{1}{4}\\left(y^{2}+2y\\right)=\\frac{115}{4}-\\frac{1}{1}\\left(25\\right)" }, { "type": "step", "primary": "Convert $$y\\:$$to square form", "result": "-\\frac{1}{1}\\left(x-5\\right)^{2}+\\frac{1}{4}\\left(y^{2}+2y+1\\right)=\\frac{115}{4}-\\frac{1}{1}\\left(25\\right)+\\frac{1}{4}\\left(1\\right)" }, { "type": "step", "primary": "Convert to square form", "result": "-\\frac{1}{1}\\left(x-5\\right)^{2}+\\frac{1}{4}\\left(y+1\\right)^{2}=\\frac{115}{4}-\\frac{1}{1}\\left(25\\right)+\\frac{1}{4}\\left(1\\right)" }, { "type": "step", "primary": "Refine $$\\frac{115}{4}-\\frac{1}{1}\\left(25\\right)+\\frac{1}{4}\\left(1\\right)$$", "result": "-\\frac{1}{1}\\left(x-5\\right)^{2}+\\frac{1}{4}\\left(y+1\\right)^{2}=4" }, { "type": "step", "primary": "Divide by $$4$$", "result": "-\\frac{\\left(x-5\\right)^{2}}{4}+\\frac{\\left(y+1\\right)^{2}}{16}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(y-\\left(-1\\right)\\right)^{2}}{4^{2}}-\\frac{\\left(x-5\\right)^{2}}{2^{2}}=1" } ], "meta": { "interimType": "Hyperbola Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(y-\\left(-1\\right)\\right)^{2}}{4^{2}}-\\frac{\\left(x-5\\right)^{2}}{2^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(5,\\:-1\\right),\\:a=4,\\:b=2" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=2(x-5)-1", "displayFormula": "y=2(x-5)-1", 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