foci 4y^2-25x^2=100

{ "query": { "display": "foci $$4y^{2}-25x^{2}=100$$", "symbolab_question": "CONIC#foci 4y^{2}-25x^{2}=100" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "foci", "default": "(0,\\sqrt{29}),(0,-\\sqrt{29})" }, "steps": { "type": "interim", "title": "Hyperbola foci given $$4y^{2}-25x^{2}=100:{\\quad}\\left(0,\\:\\sqrt{29}\\right),\\:\\left(0,\\:-\\sqrt{29}\\right)$$", "steps": [ { "type": "definition", "title": "Hyperbola Foci", "text": "For an up-down facing hyperbola, the Foci (focus points) are defined as $$\\left(h,\\:k+c\\right),\\:\\left(h,\\:k-c\\right),\\:$$<br/>where $$c=\\sqrt{a^2+b^2}$$ is the distance from the center $$\\left(h,\\:k\\right)\\:$$to a focus" }, { "type": "step", "result": "\\left(h,\\:k+c\\right),\\:\\left(h,\\:k-c\\right)" }, { "type": "step", "primary": "Calculate Hyperbola properties" }, { "type": "interim", "title": "$$4y^{2}-25x^{2}=100:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=2$$", "input": "4y^{2}-25x^{2}=100", "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 $$4y^{2}-25x^{2}=100\\:$$in the form of a standard hyperbola equation", "input": "4y^{2}-25x^{2}=100", "steps": [ { "type": "step", "primary": "Divide by coefficient of square terms: $$25$$", "result": "-x^{2}+\\frac{4}{25}y^{2}=4" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$4$$", "result": "-\\frac{1}{4}x^{2}+\\frac{1}{25}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "-\\frac{x^{2}}{4}+\\frac{y^{2}}{25}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(y-0\\right)^{2}}{5^{2}}-\\frac{\\left(x-0\\right)^{2}}{2^{2}}=1" } ], "meta": { "interimType": "Hyperbola Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(y-0\\right)^{2}}{5^{2}}-\\frac{\\left(x-0\\right)^{2}}{2^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=2" } ], "meta": { "interimType": "Hyperbola UpDown Top Title 3Eq" } }, { "type": "step", "result": "\\left(0,\\:0+c\\right),\\:\\left(0,\\:0-c\\right)" }, { "type": "step", "primary": "Compute $$c:$$" }, { "type": "interim", "title": "$$c=\\sqrt{5^{2}+2^{2}}:{\\quad}\\sqrt{29}$$", "input": "\\sqrt{5^{2}+2^{2}}", "steps": [ { "type": "step", "primary": "$$5^{2}=25$$", "result": "=\\sqrt{25+2^{2}}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{25+4}" }, { "type": "step", "primary": "Add the numbers: $$25+4=29$$", "result": "=\\sqrt{29}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BUa0Z7lbkNf0/S+xY3Q9OV1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIy2IXR5dVA8Sv5V/g5tZaxhuWGmGj3G77AwaA352mJ8vF2m7oyRFn7Q7v6gtKbJ6EialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "\\left(0,\\:0+\\sqrt{29}\\right),\\:\\left(0,\\:0-\\sqrt{29}\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(0,\\:\\sqrt{29}\\right),\\:\\left(0,\\:-\\sqrt{29}\\right)" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{5x}{2}", "displayFormula": "y=\\frac{5x}{2}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{5x}{2}", "displayFormula": "y=-\\frac{5x}{2}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{25(\\frac{x^{2}}{2^{2}}+1)}", 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