If the circles and intersect at two distinct points, then
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: C
Solution
The first circle has center (8,10) and radius r, the second has center (4,7) and radius 6.
Distance between centers = √((8-4)² + (10-7)²) = √(16+9) = 5.
For two distinct intersection points: |r - 6| < 5 < r + 6.
From 5 < r + 6: r > -1 (always true for positive r).
From |r - 6| < 5: -5 < r - 6 < 5, giving 1 < r < 11.
So the condition is 1 < r < 11.
Distance between centers = √((8-4)² + (10-7)²) = √(16+9) = 5.
For two distinct intersection points: |r - 6| < 5 < r + 6.
From 5 < r + 6: r > -1 (always true for positive r).
From |r - 6| < 5: -5 < r - 6 < 5, giving 1 < r < 11.
So the condition is 1 < r < 11.
[JEE Main 2019 · 9 Jan · Shift II]