{
"query": {
"display": "focos $$y^{2}=-12x$$",
"symbolab_question": "CONIC#foci y^{2}=-12x"
},
"solution": {
"level": "PERFORMED",
"subject": "Geometry",
"topic": "Parabola",
"subTopic": "foci",
"default": "(-3,0)"
},
"steps": {
"type": "interim",
"title": "Foco de la parábola dado $$y^{2}=-12x:{\\quad}\\left(-3,\\:0\\right)$$",
"steps": [
{
"type": "definition",
"title": "Foco de una parábola",
"text": "Una parábola es el espacio de puntos tal que la distancia a un punto (el foco) equivale a la distancia a una linea (la directriz)"
},
{
"type": "definition",
"title": "Ecuación general de la parábola",
"text": "$$4p\\left(x-h\\right)=\\left(y-k\\right)^{2}\\:$$ es la ecuación estándar de la parábola cuando esta se abre hacia la derecha, con vértice en $$\\left(h,\\:k\\right),\\:$$<br/>y longitud focal $$|p|$$"
},
{
"type": "interim",
"title": "Reescribir $$y^{2}=-12x\\:$$con la forma de la ecuación general de la parábola:$${\\quad}4\\left(-3\\right)\\left(x-0\\right)=\\left(y-0\\right)^{2}$$",
"input": "y^{2}=-12x",
"steps": [
{
"type": "step",
"primary": "Intercambiar lados",
"result": "-12x=y^{2}"
},
{
"type": "step",
"primary": "Factorizar $$4$$",
"result": "4\\cdot\\:\\frac{-12}{4}x=y^{2}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "4\\left(-3\\right)x=y^{2}"
},
{
"type": "step",
"primary": "Reescribir como",
"result": "4\\left(-3\\right)\\left(x-0\\right)=\\left(y-0\\right)^{2}"
}
],
"meta": {
"interimType": "Parabola Canonical Format Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aRgRQFW0MqhVT4GbzWEc4k7qsHRxN9bqMsP43ThlkcsWgM/Qnh8bwlZVgLtIW15FQXpHylPs0Rersq/WcBN38o9ZLTbv6820yRhHB5ziAqjzWuLWZ/TGm41cerLey75mp+53YEFIjmIWpl3qnAYzGk3kCh3oevUunZ7/b0qFKBSZs1nkTJfkQYLC0yB+pc8p7hRvB2fGDDYyDtPvP7scNiYj9X0ofiLRzFBiz0YRsOhxOOQ66Hs89Yo5i8zsDqimVwYL+6AEo+OF1DlNrXHoLLi/3UF9VzfTSW0upjWi+Iw="
}
},
{
"type": "step",
"result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-3"
},
{
"type": "step",
"primary": "La parábola es simétrica al rededor del eje x (abscisas) y, por lo tanto, el foco el foco yace a una distancia $$p$$ del centro $$\\left(0,\\:0\\right)$$ a lo largo del eje x (abscisas) ",
"result": "\\left(0+p,\\:0\\right)"
},
{
"type": "step",
"result": "=\\left(0+\\left(-3\\right),\\:0\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "\\left(-3,\\:0\\right)"
}
],
"meta": {
"solvingClass": "Parabola"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"funcsToDraw": {
"funcs": [
{
"evalFormula": "y=\\sqrt{4(-3)x}+0",
"displayFormula": "4(-3)x=y^{2}",
"attributes": {
"color": "PURPLE",
"lineType": "NORMAL",
"isAsymptote": false
}
},
{
"evalFormula": "y=-\\sqrt{4(-3)x}+0",
"displayFormula": "4(-3)x=y^{2}",
"attributes": {
"color": "PURPLE",
"lineType": "NORMAL",
"isAsymptote": false
}
},
{
"evalFormula": "x=3",
"displayFormula": "x=3",
"attributes": {
"color": "GRAY",
"lineType": "NORMAL",
"labels": [
"\\mathrm{directrix}"
],
"isAsymptote": false
}
}
]
},
"pointsToDraw": {
"pointsLatex": [
"(0,0)",
"(-3,0)"
],
"pointsDecimal": [
{
"fst": 0,
"snd": 0
},
{
"fst": -3,
"snd": 0
}
],
"attributes": [
{
"color": "PURPLE",
"labels": [
"\\mathrm{vertex}"
],
"labelTypes": [
"DEFAULT"
],
"labelColors": [
"PURPLE"
]
},
{
"color": "PURPLE",
"labels": [
"\\mathrm{focus}"
],
"labelTypes": [
"DEFAULT"
],
"labelColors": [
"PURPLE"
]
}
]
},
"functionChanges": [
{
"origFormulaLatex": [],
"finalFormulaLatex": [],
"plotTitle": "4(-3)(x)=y^{2}",
"paramsLatex": [],
"paramsReplacementsLatex": []
}
],
"localBoundingBox": {
"xMin": -33.75,
"xMax": 33.75,
"yMin": -33.75,
"yMax": 33.75
}
},
"showViewLarger": true
}
}
}
Solución
focos
Solución
Pasos de solución
Reescribir con la forma de la ecuación general de la parábola:
La parábola es simétrica al rededor del eje x (abscisas) y, por lo tanto, el foco el foco yace a una distancia del centro a lo largo del eje x (abscisas)
Simplificar