{
"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
}
}
}
Solution
foci
Solution
Solution steps
Calculate Hyperbola properties
Up-down Hyperbola with
Compute
Refine
Graph
Popular Examples
solvefor y,36x^2-25y^2+144x-50y+119=0solve for asymptotes of (x^2)/(100)-(y^2)/(64)=1asymptotes x^2+y^2<= 4vertices f(x)=x^2+4x+3vertices directrix y= 1/12 x^2directrix
Frequently Asked Questions (FAQ)
What is the foci (y^2)/(49)-(x^2)/(64)=1 ?
The foci (y^2)/(49)-(x^2)/(64)=1 is (0,sqrt(113)),(0,-sqrt(113))