解答这个问题---Problem Description: 例 1. 平面直角坐标系 $xOy$ 中，经过椭圆 C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ 的一个焦点的直线 $x - y - \sqrt{3} = 0$ 与 C 相交于 M, N 两点，P 为 MN 的中点，且 OP 斜率是 $-\frac{1}{4}$. (I) 求椭圆 C 的方程; (II) 直线 $l$ 分别与椭圆 $C$ 和圆 $D: x^2 + y^2 = r^2 (b < r < a)$ 相切于点 $A$、 $B$，求 $|AB|$ 的最大值. Analysis: (I) 点差法，略 (II) Diagram Description: Type: Coordinate plane with geometric shapes and labeled points/lines. Coordinate Axes: X-axis and Y-axis intersect at the origin O. X-axis is horizontal, labeled with positive direction to the right, scaled with ticks at -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5. Y-axis is vertical, labeled with positive direction upwards, scaled with ticks at -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2. The origin (0,0) is labeled O. Geometric Shapes: - Green curve: Appears to be an ellipse centered at the origin. It passes through approximately (2, 0), (-2, 0), (0, 1), (0, -1). - Blue curve: Appears to be a circle centered at the origin. It passes through approximately (1.5, 0), (-1.5, 0), (0, 1.5), (0, -1.5). Its radius seems to be around 1.5. - Red line: A straight line labeled 'f'. It has a positive slope and intersects both the ellipse and the circle. Points: - O: The origin (0,0), labeled O. - A: A point on the green ellipse and the red line, approximately at coordinates (-1.5, 0.5). Labeled A. - B: A point on the blue circle and the red line, approximately at coordinates (-0.8, 1.3). Labeled B. Lines/Segments: - Red line 'f': Passes through A and B. - Black segments: OA and OB are drawn as line segments from the origin O to points A and B, respectively. Relative Position and Direction: Point A is on the green ellipse and the red line. Point B is on the blue circle and the red line. The red line appears to be tangent to the ellipse at A and tangent to the circle at B, based on the context of part (II) and the visual representation of A and B on the shapes. O is the common center of the ellipse and the circle.