
### 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 x2/16 – y2/9 = 1, think step by step
### Assistant: Given the equation x2/16 – y2/9 = 1, we can compare it with the standard equation of a hyperbola x2/a2 – y2/b2 = 1. By doing so, we find that a = 4 and b = 3. We then calculate c using a2 + b2 = c2, which gives us c = 5. The coordinates of the foci are (±c, 0), so they are (±5, 0). The coordinates of the vertices are (±a, 0), so they are (±4, 0). The eccentricity is e = c/a = 5/4. The length of the latus rectum is given by 2b2/a = (2×9)/4 = 9/2. , The answer is The coordinates of the foci are (±5, 0), the coordinates of the vertices are (±4, 0), the eccentricity is e = c/a = 5/4, and the length of the latus rectum is 9/2.
