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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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