There are n stations in a circular path.Two consecutive stations are connected by blue line and two non-consecutive stations are connected by red line.If no. of red lines is equal to 99 times number of blue line then value of n is

Options

A. 201
B. 200
C. 199
D. 202

Answer

To solve the problem, we need to determine the number of blue and red lines based on the number of stations \( n \) in a circular path.

### Step 1: Calculate the number of blue lines
In a circular arrangement of \( n \) stations, each station is connected to its two consecutive neighbors by blue lines. Therefore, the number of blue lines can be calculated as:
\[ \text{Number of blue lines} = n \]

### Step 2: Calculate the number of red lines
Red lines connect two non-consecutive stations. To find the number of red lines, we need to consider how many ways we can choose two stations that are not consecutive.

1. The total number of ways to choose 2 stations from \( n \) stations is given by \( \binom{n}{2} \).
2. The number of ways to choose 2 consecutive stations is \( n \) (since each station has exactly one consecutive station on either side).
3. Therefore, the number of red lines is:
\[ \text{Number of red lines} = \binom{n}{2} - n = \frac{n(n-1)}{2} - n = \frac{n(n-1) - 2n}{2} = \frac{n(n-3)}{2} \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the number of red lines is equal to 99 times the number of blue lines:
\[ \frac{n(n-3)}{2} = 99n \]

### Step 4: Simplify the equation
To eliminate the fraction, multiply both sides by 2:
\[ n(n-3) = 198n \]
Rearranging gives:
\[ n^2 - 201n = 0 \]

### Step 5: Factor the equation
Factoring out \( n \):
\[ n(n - 201) = 0 \]
This gives us two solutions:
\[ n = 0 \quad \text{or} \quad n = 201 \]

Since \( n \) must be a positive integer representing the number of stations, we have:
\[ n = 201 \]

### Conclusion
The value of \( n \) is:
\[ \boxed{201} \]