How do I find the equation of a hyperbola with given foci and asymptotes?

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To find the equation of a hyperbola centered at the origin with given foci at (8,0) and (-8,0), the distance to the foci indicates c=8. The asymptotes are given as y=4x and y=-4x, leading to the relationship b/a=4. By using the equation c²=a²+b², it can be established that a²+b²=64. Substituting b=4a into this equation allows for the calculation of a and b, ultimately leading to the standard form of the hyperbola's equation. The discussion emphasizes the importance of solving the system of equations to derive the hyperbola's parameters.
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Homework Statement


Find an equation of the hyperbola with it's center at the origin.

Foci:(8,0),(-8,0) Asymptotes: y=4x, y=-4x

Homework Equations


Equation for the asymptotes of a hyperbola with a horizontal transverse axis
y=k\pm\frac{b}{a}(x-h)

Equation for a hyperbola centered at (0,0) and having a horizontal transverse axis in standard form

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

The Attempt at a Solution



Ok, so I need to find a and b to write the equation.

I can deduce from the information given that c=8 which is the distance from the center to a focus.

Therefore I can declare that a^2+b^2=64 and from looking at the asymptotes I can also declare that \frac{b}{a}=4. I don't know how to solve a system of equations with division in it. Is there something I am missing?
 
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Put b = 4a from the second equation in the first equation and solve for a. Then solve for b.
 
LCKurtz said:
Put b = 4a from the second equation in the first equation and solve for a. Then solve for b.
Ahh.. Yes.

Thanks!
 

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