foci (y^2)/(36)-(x^2)/(64)=1

{ "query": { "display": "foci $$\\frac{y^{2}}{36}-\\frac{x^{2}}{64}=1$$", "symbolab_question": "CONIC#foci \\frac{y^{2}}{36}-\\frac{x^{2}}{64}=1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "foci", "default": "(0,10),(0,-10)" }, "steps": { "type": "interim", "title": "Hyperbola foci given $$\\frac{y^{2}}{36}-\\frac{x^{2}}{64}=1:{\\quad}\\left(0,\\:10\\right),\\:\\left(0,\\:-10\\right)$$", "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": "$$\\frac{y^{2}}{36}-\\frac{x^{2}}{64}=1:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=6,\\:b=8$$", "input": "\\frac{y^{2}}{36}-\\frac{x^{2}}{64}=1", "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": "step", "primary": "Rewrite $$\\frac{y^{2}}{36}-\\frac{x^{2}}{64}=1\\:$$in the form of a standard hyperbola equation", "result": "\\frac{\\left(y-0\\right)^{2}}{6^{2}}-\\frac{\\left(x-0\\right)^{2}}{8^{2}}=1" }, { "type": "step", "primary": "Therefore Hyperbola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=6,\\:b=8" } ], "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": "$$c=\\sqrt{6^{2}+8^{2}}:{\\quad}10$$", "input": "\\sqrt{6^{2}+8^{2}}", "steps": [ { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36+8^{2}}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=\\sqrt{36+64}" }, { "type": "step", "primary": "Add the numbers: $$36+64=100$$", "result": "=\\sqrt{100}" }, { "type": "step", "primary": "Factor the number: $$100=10^{2}$$", "result": "=\\sqrt{10^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{10^{2}}=10$$" ], "result": "=10", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VIjqvJet4lYeIYiDko76dl1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIt9sw8DbT/PSwY9NtCaephNvJ4nZ+5mdhpj/pkQDBF2+c7hA9jbkKH2+2yKgLGz5t" } }, { "type": "step", "result": "\\left(0,\\:0+10\\right),\\:\\left(0,\\:0-10\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(0,\\:10\\right),\\:\\left(0,\\:-10\\right)" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{3x}{4}", "displayFormula": "y=\\frac{3x}{4}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{3x}{4}", "displayFormula": "y=-\\frac{3x}{4}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{36(\\frac{x^{2}}{8^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{6^{2}}-\\frac{x^{2}}{8^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{36(\\frac{x^{2}}{8^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{6^{2}}-\\frac{x^{2}}{8^{2}}=1", "attributes": { "color": "PURPLE", "lineType": 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