The standard equation of a vertical hyperbola with center (h, k), transverse axis 2a, and conjugate axis 2b is:
((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1
Here, the center is (6, 4), the transverse axis is 6, and the conjugate axis is 10.
Since this is a vertical hyperbola, the transverse axis is the axis that is parallel to the y-axis. Therefore, a = 3.
Similarly, the conjugate axis is the axis that is parallel to the x-axis. Therefore, b = 5.
The center is (6, 4), so h = 6 and k = 4.
Substituting these values in the standard equation, we get:
((y - 4)^2 / 3^2) - ((x - 6)^2 / 5^2) = 1
Simplifying, we get:
((y - 4)^2 / 9) - ((x - 6)^2 / 25) = 1
Therefore, the equation of the vertical hyperbola with center at (6, 4), a transverse axis of 6, and a conjugate axis of 10 is:
((y - 4)^2 / 9) - ((x - 6)^2 / 25) = 1