You are given `n ( ge 3)` circles having different radical axes and radical centres. The value of 'n' for which the number of radical axes is equal to the number of radical centres is

Options

A. 3
B. 4
C. 5
D. 8

Answer

To solve the problem, we need to find the value of \( n \) (where \( n \geq 3 \)) such that the number of radical axes is equal to the number of radical centers for \( n \) circles.

### Step-by-Step Solution:

1. **Understanding Radical Axes and Centers**:
- The radical axis of two circles is defined as the locus of points that have equal power with respect to both circles.
- The radical center of three circles is the point where the radical axes of each pair of circles intersect.

2. **Finding the Number of Radical Axes**:
- The number of radical axes formed by \( n \) circles can be calculated using combinations. Specifically, the number of radical axes is given by \( \binom{n}{2} \), which represents the number of ways to choose 2 circles from \( n \) circles.

3. **Finding the Number of Radical Centers**:
- The number of radical centers formed by \( n \) circles is given by \( \binom{n}{3} \), which represents the number of ways to choose 3 circles from \( n \) circles.

4. **Setting Up the Equation**:
- We need to find \( n \) such that:
\[ \binom{n}{2} = \binom{n}{3} \]

5. **Calculating the Combinations**:
- The combination formulas are:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
\[ \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \]

6. **Equating the Two Expressions**:
- Setting the two expressions equal gives us:
\[ \frac{n(n-1)}{2} = \frac{n(n-1)(n-2)}{6} \]

7. **Cross-Multiplying**:
- Cross-multiplying to eliminate the fractions:
\[ 6n(n-1) = 2n(n-1)(n-2) \]

8. **Simplifying the Equation**:
- Dividing both sides by \( 2n(n-1) \) (assuming \( n \neq 0 \) and \( n \neq 1 \)):
\[ 3 = n - 2 \]

9. **Solving for \( n \)**:
- Rearranging gives:
\[ n = 5 \]

10. **Conclusion**:
- Therefore, the value of \( n \) for which the number of radical axes is equal to the number of radical centers is \( n = 5 \).

### Final Answer:
The value of \( n \) is **5**.