A man running round a racecourse notes that the sum of the distance of two flag-posts from him is always 10 m and the distance between the flag posts is 8m. The area of the path he encloses is:

Options

A. `18pi` sq m
B. `15pi` sq m
C. `12pi` sq m
D. `8pi` sq m

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the Problem
The problem states that a man running around a racecourse finds that the sum of the distances from two flag-posts (let's call them FP1 and FP2) is always 10 meters, and the distance between the two flag-posts is 8 meters. We need to find the area of the path he encloses.

### Step 2: Identify the Shape
The condition that the sum of the distances from two points (the flag-posts) is constant indicates that the path forms an ellipse. The two flag-posts are the foci of the ellipse.

### Step 3: Determine the Values of 'a' and 'c'
1. The sum of the distances from the foci (flag-posts) to any point on the ellipse is equal to 10 meters. This means:
\[ 2a = 10 \implies a = 5 \text{ meters} \]

2. The distance between the two flag-posts (the foci) is 8 meters, which gives:
\[ 2c = 8 \implies c = 4 \text{ meters} \]

### Step 4: Use the Relationship Between a, b, and c
In an ellipse, the relationship between the semi-major axis (a), semi-minor axis (b), and the distance to the foci (c) is given by the equation:
\[ c^2 = a^2 - b^2 \]
Substituting the known values:
\[ 4^2 = 5^2 - b^2 \]
\[ 16 = 25 - b^2 \]
\[ b^2 = 25 - 16 = 9 \implies b = 3 \text{ meters} \]

### Step 5: Calculate the Area of the Ellipse
The area \( A \) of an ellipse is given by the formula:
\[ A = \pi \cdot a \cdot b \]
Substituting the values of \( a \) and \( b \):
\[ A = \pi \cdot 5 \cdot 3 = 15\pi \text{ square meters} \]

### Step 6: Conclusion
The area of the path enclosed by the man running around the racecourse is:
\[ \boxed{15\pi} \text{ square meters} \]