Write the equation in standard form of the hyperbola with center at the origin, its transverse axis is vertical, and it's asymptotes are y=+/- (8/5)x
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To determine the equation, apply the formula of hyperbola with vertical transverse axis which is:
`(y-k)^2/a^2-(x-h)^2/b^2=1`
Since the center is (0,0), plug-in h=0 and k=0.
`(y-0)^2/a^2-(x-0)^2/b^2=1`
`y^2/a^2-x^2/b^2=1`
To solve for the values of a and b, apply the formula of asymptotes of hyperbola with vertical transverse axis.
`y=+-a/b(x-h)+k`
Again, plug-in the values of h and k.
`y=+-a/b(x-0)+0`
`y=+-a/bx`
Then, substitute `y=+-8/5x` .
`+-8/5x=+-a/bx`
To simplify this, divide both sides by x.
`(+-8/5x)/x=(+-a/bx)/x`
`+-8/5=+-a/b`
So, a=8 and b=5.
Now that the values of a and b are known, plug these to:
`y^2/a^2-x^2/b^2=1`
`y^2/8^2-x^2/5^2=1`
`y^2/64-x^2/25=1`
Hence, the equation of the hyperbola is `y^2/64-x^2/25=1` .
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