{
"query": {
"display": "focos $$16x^{2}+25y^{2}=400$$",
"symbolab_question": "CONIC#foci 16x^{2}+25y^{2}=400"
},
"solution": {
"level": "PERFORMED",
"subject": "Geometry",
"topic": "Ellipse",
"subTopic": "foci",
"default": "(3,0),(-3,0)"
},
"steps": {
"type": "interim",
"title": "Focos de la elipse dados $$16x^{2}+25y^{2}=400:{\\quad}\\left(3,\\:0\\right),\\:\\left(-3,\\:0\\right)$$",
"steps": [
{
"type": "definition",
"title": "Focos de una elipse",
"text": "Para una elipse con un eje mayor paralelo al eje x (abscisas), los focos se definen como $$\\left(h+c,\\:k\\right),\\:\\left(h-c,\\:k\\right),\\:$$<br/> donde $$c=\\sqrt{a^2-b^2}\\:$$ es la distancia del centro $$\\left(h,\\:k\\right)\\:$$a uno de los focos"
},
{
"type": "step",
"result": "\\left(h+c,\\:k\\right),\\:\\left(h-c,\\:k\\right)"
},
{
"type": "step",
"primary": "Calcular las propiedades de la elipse"
},
{
"type": "interim",
"title": "$$16x^{2}+25y^{2}=400:{\\quad}$$Elipse con centro $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=4$$",
"input": "16x^{2}+25y^{2}=400",
"steps": [
{
"type": "definition",
"title": "Elipse con centro fuera del origen",
"text": "$$\\frac{\\left(x-h\\right)^{2}}{a^2}+\\frac{\\left(y-k\\right)^{2}}{b^2}=1\\:$$es la ecuación de la elipse con centro fuera del origen<br/>con centro en $$\\left(h,\\:k\\right)\\:$$y $$a,\\:b$$ son los semiejes mayor y menor"
},
{
"type": "interim",
"title": "Reescribir $$16x^{2}+25y^{2}=400\\:$$con la forma de la ecuación general de la elipse",
"input": "16x^{2}+25y^{2}=400",
"steps": [
{
"type": "step",
"primary": "Dividir entre el coeficiente de términos cuadrados: $$16$$",
"result": "x^{2}+\\frac{25}{16}y^{2}=25"
},
{
"type": "step",
"primary": "Dividir entre el coeficiente de términos cuadrados: $$25$$",
"result": "\\frac{1}{25}x^{2}+\\frac{1}{16}y^{2}=1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "\\frac{x^{2}}{25}+\\frac{y^{2}}{16}=1"
},
{
"type": "step",
"primary": "Reescribir en la forma estándar",
"result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}+\\frac{\\left(y-0\\right)^{2}}{4^{2}}=1"
}
],
"meta": {
"interimType": "Ellipse Canonical Format 1Eq"
}
},
{
"type": "step",
"result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}+\\frac{\\left(y-0\\right)^{2}}{4^{2}}=1"
},
{
"type": "step",
"primary": "Por lo tanto, las propiedades de la elipse son:",
"result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=4"
},
{
"type": "step",
"primary": "$$a>b\\:$$por lo tanto $$a\\:$$es un semieje mayor y $$b\\:$$es un semieje menor",
"result": "\\mathrm{Elipse\\:con\\:centro}\\:\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=4"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "\\left(0+c,\\:0\\right),\\:\\left(0-c,\\:0\\right)"
},
{
"type": "step",
"primary": "Calcular $$c:$$"
},
{
"type": "interim",
"title": "$$c=\\sqrt{5^{2}-4^{2}}:{\\quad}3$$",
"input": "\\sqrt{5^{2}-4^{2}}",
"steps": [
{
"type": "step",
"primary": "$$5^{2}=25$$",
"result": "=\\sqrt{25-4^{2}}"
},
{
"type": "step",
"primary": "$$4^{2}=16$$",
"result": "=\\sqrt{25-16}"
},
{
"type": "step",
"primary": "Restar: $$25-16=9$$",
"result": "=\\sqrt{9}"
},
{
"type": "step",
"primary": "Descomponer el número en factores primos: $$9=3^{2}$$",
"result": "=\\sqrt{3^{2}}"
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt[n]{a^n}=a$$",
"secondary": [
"$$\\sqrt{3^{2}}=3$$"
],
"result": "=3",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74f6cUYKncTCZw7Lf+vpnnl1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIFNZvSS5A2llKcsOioUT3tE+QZDw8sNbXVMhDKVB/Cmpw8B6izv73gtWusAUWvvnM"
}
},
{
"type": "step",
"result": "\\left(0+3,\\:0\\right),\\:\\left(0-3,\\:0\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "\\left(3,\\:0\\right),\\:\\left(-3,\\:0\\right)"
}
],
"meta": {
"solvingClass": "Ellipse"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"funcsToDraw": {
"funcs": [
{
"evalFormula": "y=\\sqrt{16(1-\\frac{x^{2}}{5^{2}})}",
"displayFormula": "\\frac{x^{2}}{5^{2}}+\\frac{y^{2}}{4^{2}}=1",
"attributes": {
"color": "PURPLE",
"lineType": "NORMAL",
"isAsymptote": false
}
},
{
"evalFormula": "y=-\\sqrt{16(1-\\frac{x^{2}}{5^{2}})}",
"displayFormula": "\\frac{x^{2}}{5^{2}}+\\frac{y^{2}}{4^{2}}=1",
"attributes": {
"color": "PURPLE",
"lineType": "NORMAL",
"isAsymptote": false
}
}
]
},
"pointsToDraw": {
"pointsLatex": [
"(0,0)",
"(3,0)",
"(-3,0)"
],
"pointsDecimal": [
{
"fst": 0,
"snd": 0
},
{
"fst": 3,
"snd": 0
},
{
"fst": -3,
"snd": 0
}
],
"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": "5",
"p2y": "0",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"labels": [
"a=5"
],
"isAsymptote": false
}
},
{
"p1x": "0",
"p1y": "0",
"p2x": "0",
"p2y": "4",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"labels": [
"b=4"
],
"isAsymptote": false
}
}
],
"functionChanges": [
{
"origFormulaLatex": [],
"finalFormulaLatex": [],
"plotTitle": "\\frac{x^{2}}{5^{2}}+\\frac{y^{2}}{4^{2}}=1",
"paramsLatex": [],
"paramsReplacementsLatex": []
}
],
"localBoundingBox": {
"xMin": -11.25,
"xMax": 11.25,
"yMin": -11.25,
"yMax": 11.25
}
},
"showViewLarger": true
}
}
}
Solución
focos
Solución
Pasos de solución
Calcular las propiedades de la elipse
Elipse con centro
Calcular
Simplificar