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The line y = mx + 1 cuts through the circle (x - 3)² + (y + 2)² = 25 at two points, P and Q. If the x-coordinate of the midpoint of segment PQ is -3/5, which of the following is true about the range of m?
- -3 ≤ m < -1
- 2 ≤ m < 4
- 4 ≤ m < 6
- 6 ≤ m < 8
Correct answer: 2 ≤ m < 4
Solution
The chord midpoint is the foot of the perpendicular from center (3,-2) to y=mx+1, with x = 3(1-m)/(m^2+1). Setting this equal to -3/5 gives m=2 or m=3, both lying in 2 <= m < 4 (index 1). The stored range 4<=m<6 is incorrect.
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