{ "query": { "display": "foci $$\\frac{y^{2}}{49}-\\frac{x^{2}}{64}=1$$", "symbolab_question": "CONIC#foci \\frac{y^{2}}{49}-\\frac{x^{2}}{64}=1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Hyperbola", "subTopic": "foci", "default": "(0,\\sqrt{113}),(0,-\\sqrt{113})" }, "steps": { "type": "interim", "title": "Hyperbola foci given $$\\frac{y^{2}}{49}-\\frac{x^{2}}{64}=1:{\\quad}\\left(0,\\:\\sqrt{113}\\right),\\:\\left(0,\\:-\\sqrt{113}\\right)$$", "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": "$$\\frac{y^{2}}{49}-\\frac{x^{2}}{64}=1:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=7,\\:b=8$$", "input": "\\frac{y^{2}}{49}-\\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 $$\\frac{y^{2}}{49}-\\frac{x^{2}}{64}=1\\:$$in the form of a standard hyperbola equation", "result": "\\frac{\\left(y-0\\right)^{2}}{7^{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=7,\\: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": "$$c=\\sqrt{7^{2}+8^{2}}:{\\quad}\\sqrt{113}$$", "input": "\\sqrt{7^{2}+8^{2}}", "steps": [ { "type": "step", "primary": "$$7^{2}=49$$", "result": "=\\sqrt{49+8^{2}}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=\\sqrt{49+64}" }, { "type": "step", "primary": "Add the numbers: $$49+64=113$$", "result": "=\\sqrt{113}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A8YDEosbx1vf0Ca6pq7mBF1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIryiL9ze4LkaXvgKKJlOABV7YqADjK16DLU2UXrHp1Y3rW4db1A1GCRxwkf4N+jAUFkVDw5yEF2DEw4msQzKTew==" } }, { "type": "step", "result": "\\left(0,\\:0+\\sqrt{113}\\right),\\:\\left(0,\\:0-\\sqrt{113}\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(0,\\:\\sqrt{113}\\right),\\:\\left(0,\\:-\\sqrt{113}\\right)" } ], "meta": { "solvingClass": "Hyperbola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{7x}{8}", "displayFormula": "y=\\frac{7x}{8}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=-\\frac{7x}{8}", "displayFormula": "y=-\\frac{7x}{8}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": true } }, { "evalFormula": "y=\\sqrt{49(\\frac{x^{2}}{8^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{7^{2}}-\\frac{x^{2}}{8^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{49(\\frac{x^{2}}{8^{2}}+1)}", "displayFormula": "\\frac{y^{2}}{7^{2}}-\\frac{x^{2}}{8^{2}}=1", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(0,\\sqrt{113})", "(0,-\\sqrt{113})" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0, "snd": 10.63014581273465 }, { "fst": 0, "snd": -10.63014581273465 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{Center}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "linesToDraw": [ { "p1x": "0", "p1y": "0", "p2x": "0", "p2y": "7", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "a=7" ], "isAsymptote": false } }, { "p1x": "0", "p1y": "0", "p2x": "8", "p2y": "0", "attributes": { "color": "GRAY", "lineType": "BOLD", "labels": [ "b=8" ], "isAsymptote": false } } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "\\frac{y^{2}}{7^{2}}-\\frac{x^{2}}{8^{2}}=1", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -36, "xMax": 36, "yMin": -36, "yMax": 36 } }, "showViewLarger": true } } }