how to sove this qusetion in english---Question Number: 7. Question Stem: 若圆 $x^2+(y+2)^2=r^2(r>0)$ 上到直线 $y=\sqrt{3}x+2$ 的距离为1的点有且仅有2个, 则 $r$ 的取值范围是 Options: A. (0,1) B. (1,3) C. (3,+∞) D. (0,+∞)

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Let's analyze this step by step. We have a circle with center at (0, -2) and radius r. We need to find when exactly 2 points on this circle are at distance 1 from the line y equals square root of 3 times x plus 2. The key insight is that points at distance 1 from a line form two parallel lines on either side of the original line. To find points at distance 1 from the line, we first convert the line equation to standard form. The line y equals square root of 3 times x plus 2 becomes square root of 3 times x minus y plus 2 equals 0. Points at distance 1 satisfy the distance formula, giving us the absolute value equation. This creates two parallel lines: L1 where square root of 3 times x minus y equals 0, and L2 where square root of 3 times x minus y plus 4 equals 0. Now we calculate the distances from the circle center to each parallel line. The distance from center C at (0, -2) to line L1 is 1, and to line L2 is 3. This is crucial because the number of intersections between a circle and a line depends on comparing the radius r with the distance d. If r is greater than d, we get 2 intersections. If r equals d, we get 1 intersection. If r is less than d, we get 0 intersections. Let's analyze what happens for different values of r. When r is between 0 and 1, the circle is too small to intersect either line, giving 0 total intersections. When r equals 1, the circle just touches L1, giving 1 intersection. When r is between 1 and 3, the circle intersects L1 at 2 points but doesn't reach L2, giving exactly 2 intersections. When r equals 3, we get 3 intersections, and when r is greater than 3, we get 4 intersections. Therefore, for exactly 2 intersections, r must be in the interval (1, 3). Let's summarize our solution. We have a circle with center at (0, -2) and radius r. The line y equals square root of 3 times x plus 2 creates two parallel lines at distance 1: L1 at distance 1 from the center, and L2 at distance 3. For exactly 2 intersection points, the circle must intersect L1 twice and L2 zero times. This happens when the radius is greater than 1 but less than 3. Therefore, the answer is r in the interval (1, 3), which corresponds to option B.