
### Instruction: help me answer this question Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola y2/9 – x2/27 = 1, think step by step
### Assistant: Given the equation y2/9 – x2/27 = 1, we can compare it with the standard equation of a hyperbola y2/a2 – x2/b2 = 1. By doing so, we find that a = 3 and b = √27. We then calculate c using a2 + b2 = c2, which gives us c = 6. The coordinates of the foci are (0, ±c), so they are (0, 6) and (0, -6). The coordinates of the vertices are (0, ±a), so they are (0, 3) and (0, -3). The eccentricity is e = c/a = 6/3 = 2. The length of the latus rectum is given by 2b2/a = (2 × 27)/3 = 18. , The answer is The coordinates of the foci are (0, 6) and (0, -6), the coordinates of the vertices are (0, 3) and (0, -3), the eccentricity is e = c/a = 2, and the length of the latus rectum is 18.
