{
"query": {
"display": "$$9y^{2}-16x^{2}=144$$",
"symbolab_question": "CONIC#9y^{2}-16x^{2}=144"
},
"solution": {
"level": "PERFORMED",
"subject": "Geometry",
"topic": "Hyperbola",
"subTopic": "formula",
"default": "(h,k)=(0,0),a=4,b=3"
},
"steps": {
"type": "interim",
"title": "$$9y^{2}-16x^{2}=144:\\quad$$Hipérbola que abre hacia arriba y hacia abajo con $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=4,\\:b=3$$",
"input": "9y^{2}-16x^{2}=144",
"steps": [
{
"type": "definition",
"title": "Ecuación normal de la hipérbola",
"text": "$$\\frac{\\left(y-k\\right)^{2}}{a^2}-\\frac{\\left(x-h\\right)^{2}}{b^2}=1\\:$$ es la ecuación normal de la hipérbola cuando esta abre hacia arriba y hacia abajo<br/>con centro $$\\bold{\\left(h,\\:k\\right)},\\:$$ semieje $$\\bold{a}\\:$$y semieje conjugado $$\\bold{b}.$$"
},
{
"type": "interim",
"title": "Reescribir $$9y^{2}-16x^{2}=144\\:$$con la forma de la ecuación general de la hipérbola",
"input": "9y^{2}-16x^{2}=144",
"steps": [
{
"type": "step",
"primary": "Dividir entre el coeficiente de términos cuadrados: $$16$$",
"result": "-x^{2}+\\frac{9}{16}y^{2}=9"
},
{
"type": "step",
"primary": "Dividir entre el coeficiente de términos cuadrados: $$9$$",
"result": "-\\frac{1}{9}x^{2}+\\frac{1}{16}y^{2}=1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "-\\frac{x^{2}}{9}+\\frac{y^{2}}{16}=1"
},
{
"type": "step",
"primary": "Reescribir en la forma estándar",
"result": "\\frac{\\left(y-0\\right)^{2}}{4^{2}}-\\frac{\\left(x-0\\right)^{2}}{3^{2}}=1"
}
],
"meta": {
"interimType": "Hyperbola Canonical Format 1Eq"
}
},
{
"type": "step",
"result": "\\frac{\\left(y-0\\right)^{2}}{4^{2}}-\\frac{\\left(x-0\\right)^{2}}{3^{2}}=1"
},
{
"type": "step",
"primary": "Por lo tanto, las propiedades de la hipérbola son: ",
"result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=4,\\:b=3"
}
],
"meta": {
"solvingClass": "Hyperbola"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"funcsToDraw": {
"funcs": [
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"evalFormula": "y=\\frac{4x}{3}",
"displayFormula": "y=\\frac{4x}{3}",
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"attributes": {
"color": "PURPLE",
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},
{
"evalFormula": "y=\\sqrt{16(\\frac{x^{2}}{3^{2}}+1)}",
"displayFormula": "\\frac{y^{2}}{4^{2}}-\\frac{x^{2}}{3^{2}}=1",
"attributes": {
"color": "PURPLE",
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{
"evalFormula": "y=-\\sqrt{16(\\frac{x^{2}}{3^{2}}+1)}",
"displayFormula": "\\frac{y^{2}}{4^{2}}-\\frac{x^{2}}{3^{2}}=1",
"attributes": {
"color": "PURPLE",
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"pointsDecimal": [
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"snd": 0
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"attributes": [
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"color": "PURPLE",
"labels": [
"\\mathrm{Center}"
],
"labelTypes": [
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"labelColors": [
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},
"linesToDraw": [
{
"p1x": "0",
"p1y": "0",
"p2x": "0",
"p2y": "4",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"labels": [
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"isAsymptote": false
}
},
{
"p1x": "0",
"p1y": "0",
"p2x": "3",
"p2y": "0",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"labels": [
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"functionChanges": [
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"plotTitle": "\\frac{y^{2}}{4^{2}}-\\frac{x^{2}}{3^{2}}=1",
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"paramsReplacementsLatex": []
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"xMax": 18,
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Solución
Solución
Pasos de solución
Reescribir con la forma de la ecuación general de la hipérbola
Por lo tanto, las propiedades de la hipérbola son: