根据图片内容生成这道题的教学视频，同时给出类似题目的解题思路---**Question Stem:** 已知直线 y=mx 与函数 f(x) = √(-x² + 8x - 12) + 1 的图象有两个交点, 则实数 m 的取值范围为 **Options:** A. [1/2, 1) B. [1/2, 4/5) C. (√13-2/6, 2] D. [1/2, 2 + √13/6) **Solution/Explanation (解析):** 由 f(x) = √(-x² + 8x - 12) + 1, 得 f(x) - 1 = √(-x² + 8x - 12). 两边平方得 [f(x) - 1]² = -x² + 8x - 12. 整理得 x² - 8x + 12 + [f(x) - 1]² = 0. 即 (x - 4)² + [f(x) - 1]² = 4 (其中 f(x) ≥ 1), 所以 f(x) 表示以 (4, 1) 为圆心, 2 为半径的圆在直线 y = 1 上方的部分; 直线 y = mx 表示过原点的直线, 如图所示. 当直线 y=mx 与 f(x) 的图象相切时, |4m - 1| / √(m² + 1) = 2, 解得 m = (2 + √13) / 6 或 m = (2 - √13) / 6 (舍去). **Chart Description:** * **Type:** Coordinate plane with a curve (arc) and lines. * **Coordinate Axes:** X-axis and Y-axis intersecting at the origin O. Positive directions are indicated by arrows. * **Origin:** Labeled as O. * **Points:** A point on the x-axis is labeled '2'. The origin is O. * **Lines:** * A horizontal dashed line labeled y=1. It intersects the x-axis at an implied point (0,1) on the Y-axis scale, and extends horizontally. Visually, the arc appears to intersect this line at two points, one at x=2 and one at x=6 (implied by the distance from x=2 and the center (4,1)). * A curve or arc is shown above the line y=1. This arc represents the graph of f(x). It starts at x=2 on the line y=1, curves upwards, and ends at an implied x=6 on the line y=1. This arc is described as part of a circle with center (4,1) and radius 2, above y=1. * A straight line passing through the origin O, labeled y=mx. This line is drawn such that it intersects the arc. Another line passing through the origin is shown tangent to the arc at some point. * **Labels:** O, y, x, y=1, y=mx. * **Relative Position:** The arc is above the line y=1. The line y=mx passes through the origin (0,0). **Other Relevant Text:** * Header: 总例 (General Example) * Header: 典型引路 (Typical Example/Guidance)