Final answer:
The asymptotes of the hyperbola given by the equation 1296y²-x²=81 are the lines y = ±(1/144)x, found by putting the equation in standard form and using the asymptotes formula for hyperbolas centered at the origin.
Step-by-step explanation:
Asymptotes of the Hyperbola
To find the asymptotes of the hyperbola given by the equation 1296y²-x²=81, we first rewrite the equation in its standard form. The equation of a hyperbola in standard form is α(y-k)² - β(x-h)² = 1, where (h,k) is the center of the hyperbola. By dividing the given equation by 81, we get y²/0.0625 - x²/1296 = 1. This standard form reveals that the hyperbola is centered at the origin (0,0) and opens up and down along the y-axis.
The asymptotes of a hyperbola are the lines that the hyperbola approaches as it extends to infinity. The general formula for the asymptotes of a hyperbola centered at the origin is y = ±(b/a)x, where a and b are the square roots of the denominators in the standard form equation of the hyperbola. For our equation, a = √1296 = 36 and b = √0.0625 = 0.25. Therefore, the equations of the asymptotes are y = ±(0.25/36)x, which simplifies to y = ±(1/144)x.