解题---```plain text 12. 已知 b 是 a, c 的等差中项, 直线 ax + by + c = 0 与圆 x² + y² + 4y - 1 = 0 交于 A, B 两点, 则 |AB| 的最小值为 ( ) Options: A. 2 B. 3 C. 4 D. 2√5 Relevant Information and Mathematical Expressions: - Condition: b 是 a, c 的等差中项, which means 2b = a + c. - Line equation: ax + by + c = 0. - Circle equation: x² + y² + 4y - 1 = 0. This can be rewritten as x² + (y + 2)² = 5. - Intersection points: A and B. - Quantity to minimize: |AB|, the distance between points A and B. Analysis based on the information: - The circle has center C(0, -2) and radius r = √5. - From the condition 2b = a + c, substitute c = 2b - a into the line equation: ax + by + (2b - a) = 0, which can be rearranged as a(x - 1) + b(y + 2) = 0. This shows that the line always passes through the point P(1, -2). - The distance from the center C(0, -2) to the fixed point P(1, -2) is CP = √((1 - 0)² + (-2 - (-2))²) = √(1² + 0²) = 1. - Let d be the distance from the center C to the line ax + by + c = 0. The length of the chord |AB| is related to the radius r and distance d by the formula |AB| = 2√(r² - d²). - Since the line passes through P, the distance d from C to the line is at most the distance from C to P, i.e., d ≤ CP = 1. - To minimize |AB| = 2√(5 - d²), we need to maximize d. The maximum possible value of d is 1, which occurs when the line is perpendicular to the segment CP at P. This line is x = 1. - The minimum value of |AB| is 2√(5 - d_max²) = 2√(5 - 1²) = 2√4 = 4. ```