If the two circles and intersect in two distinct points, then
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: A
Solution
The second circle has center (4, -1) and radius √(16+1-8) = 3. The first circle has center (1, 3) and radius r. The distance between centers is √((4-1)² + (-1-3)²) = √(9+16) = 5.
For two circles to intersect in two distinct points, the distance between centers must satisfy |r₁ - r₂| < d < r₁ + r₂, i.e., |r - 3| < 5 < r + 3. From r + 3 > 5 → r > 2, and |r - 3| < 5 → -5 < r - 3 < 5 → r < 8 (since r > 0). Combining gives 2 < r < 8.
For two circles to intersect in two distinct points, the distance between centers must satisfy |r₁ - r₂| < d < r₁ + r₂, i.e., |r - 3| < 5 < r + 3. From r + 3 > 5 → r > 2, and |r - 3| < 5 → -5 < r - 3 < 5 → r < 8 (since r > 0). Combining gives 2 < r < 8.
[1989]