Find the standard form of the equation of the hyperbola satisfying the given conditions.
Endpoints of transverse axis: (0,-8), (0,8)
Asymptote: y = 4x
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The standard form of the equation of a hyperbola with center at the origin and transverse axis along the y-axis is:
((y-k)² / a²) - ((x-h)² / b²) = 1
where (h, k) is the center of the hyperbola and "a" and "b" are the distances from the center to the vertices and foci, respectively.
From the given information, we have:
Center = (0, 0)
Vertices: (0, -8), (0, 8)
Since the transverse axis is along the y-axis, "a" is the distance from the center to the vertices, which is 8.
The equation of the asymptote is y = 4x. The slope of the asymptote is 4, so "b/a" = 4. Since "a" is 8, we can determine "b" as:
b = 4a
b = 4(8)
b = 32
Plugging these values into the standard equation, we have:
((y-0)² / 8²) - ((x-0)² / 32²) = 1
Simplifying, we get:
y² / 64 - x² / 1024 = 1
Therefore, the standard form of the equation of the hyperbola is:
y² / 64 - x² / 1024 = 1
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