{
"query": {
"display": "asymptotes $$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1$$",
"symbolab_question": "CONIC#asymptotes \\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1"
},
"solution": {
"level": "PERFORMED",
"subject": "Geometry",
"topic": "Hyperbola",
"subTopic": "asymptotes",
"default": "y=\\frac{3x}{4},\\quad y=-\\frac{3x}{4}"
},
"steps": {
"type": "interim",
"title": "Hyperbola asymptotes given $$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1:{\\quad}y=\\frac{3x}{4},\\:\\quad\\:y=-\\frac{3x}{4}$$",
"steps": [
{
"type": "definition",
"title": "Hyperbola asymptotes",
"text": "The asymptotes are the lines the hyperbola tends to at $$\\pm\\infty$$<br/>For up-down hyperbola the asymptotes are $$y=\\pm\\frac{a}{b}\\left(x-h\\right)+k$$"
},
{
"type": "step",
"result": "y=\\pm\\:\\frac{a}{b}\\left(x-h\\right)+k"
},
{
"type": "step",
"primary": "Calculate Hyperbola properties"
},
{
"type": "interim",
"title": "$$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1:\\quad$$Up-down Hyperbola with $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=3,\\:b=4$$",
"input": "\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=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 $$\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1\\:$$in the form of a standard hyperbola equation",
"result": "\\frac{\\left(y-0\\right)^{2}}{3^{2}}-\\frac{\\left(x-0\\right)^{2}}{4^{2}}=1"
},
{
"type": "step",
"primary": "Therefore Hyperbola properties are:",
"result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=3,\\:b=4"
}
],
"meta": {
"interimType": "Hyperbola UpDown Top Title 3Eq"
}
},
{
"type": "step",
"result": "y=\\frac{3}{4}\\left(x-0\\right)+0,\\:\\quad\\:y=-\\frac{3}{4}\\left(x-0\\right)+0"
},
{
"type": "step",
"primary": "Refine",
"result": "y=\\frac{3x}{4},\\:\\quad\\:y=-\\frac{3x}{4}"
}
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Solution
asymptotes
Solution
Solution steps
Calculate Hyperbola properties
Up-down Hyperbola with
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Frequently Asked Questions (FAQ)
What is the asymptotes of (y^2)/9-(x^2)/(16)=1 ?
The asymptotes of (y^2)/9-(x^2)/(16)=1 is y=(3x)/4 ,\quad y=-(3x)/4