Question

Let \( A(1,2) \) be the centre and 3 be the radius of a circle \( S \). Let \( B(-1, -1) \) be the centre and \( r \) be the radius of another circle \( S_1 \). If \( \frac{\pi}{3} \) is the angle between the circles \( S \) and \( S_1 \), then the number of possible values of \( r \) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For two circles with a known angle between them, there can be two possible values for the radius of one circle depending on the configuration.

Answer 1

The Correct Option is B

We are given the centers and radii of two circles \( S \) and \( S_1 \). The angle between the two circles is \( \frac{\pi}{3} \). We need to find the number of possible values for the radius \( r \) of the second circle. From the geometry of the problem and applying the angle between two circles, there are exactly two possible values for the radius \( r \). Thus, the correct answer is option (2), 2.