Final answer:
The equation of the hyperbola centered at the origin with a horizontal transverse axis of 12 and passing through the point (10, -8) is x^2/36 - y^2/64 = 1.
Step-by-step explanation:
The question asks to write the equation of a hyperbola centered at the origin. The hyperbola has a horizontal transverse axis of length 12 and passes through the point (10, -8). The general form of a hyperbola with a horizontal transverse axis is ∛ x^2/a^2 - y^2/b^2 = 1, where 2a is the length of the transverse axis. From the given information, a = 6. To find the value of b, we use the fact that the hyperbola passes through the point (10, -8). Plugging the coordinates into the hyperbola's equation gives us 10^2/6^2 - (-8)^2/b^2 = 1. Solving for b, we find that b = ∛ 64. Therefore, the equation of the hyperbola is ∛ x^2/36 - y^2/64 = 1.