Two points are taken at random on the given straight line segment of length a. The probability for the distance between them to exceed a given length c, where `0 lt c lt a`, is

Options

A. `(b)/(a)`
B. `(b^2)/(a^2)`
C. `((a-b)/(a))^2`
D. `((a-2b)/(a-b))^2`

Answer

To solve the problem of finding the probability that the distance between two randomly chosen points on a line segment of length \( a \) exceeds a given length \( c \) (where \( 0 < c < a \)), we can follow these steps:

### Step 1: Define the Problem
Let \( x \) and \( y \) be the two points chosen randomly on the line segment from \( 0 \) to \( a \). We need to find the probability that the distance \( |x - y| \) exceeds \( c \).

### Step 2: Set Up the Conditions
The condition \( |x - y| > c \) can be broken down into two cases:
1. \( x - y > c \)
2. \( y - x > c \)

This can be rewritten as:
- \( x > y + c \)
- \( y > x + c \)

### Step 3: Visualize the Problem
We can visualize the points \( (x, y) \) in a coordinate system where both \( x \) and \( y \) range from \( 0 \) to \( a \). This creates a square with vertices at \( (0, 0) \), \( (a, 0) \), \( (0, a) \), and \( (a, a) \).

### Step 4: Identify the Areas
The total area of the square is \( a^2 \).

Next, we need to find the area where \( |x - y| \leq c \). This area is bounded by the lines:
- \( y = x - c \)
- \( y = x + c \)

### Step 5: Determine the Area of Interest
The lines \( y = x - c \) and \( y = x + c \) intersect the boundaries of the square. We need to find the area of the region where \( |x - y| \leq c \).

1. The line \( y = x - c \) intersects the x-axis at \( (c, 0) \) and the line \( y = a \) at \( (a, a - c) \).
2. The line \( y = x + c \) intersects the y-axis at \( (0, c) \) and the line \( x = a \) at \( (a - c, a) \).

The area between these two lines forms a band around the line \( y = x \).

### Step 6: Calculate the Area of the Band
The area of the band (where \( |x - y| \leq c \)) can be calculated as follows:
- The area of the triangle formed by the points \( (0, c) \), \( (c, 0) \), and \( (0, 0) \) is \( \frac{1}{2} \times c \times c = \frac{c^2}{2} \).
- The area of the triangle formed by the points \( (a, a - c) \), \( (a - c, a) \), and \( (a, a) \) is also \( \frac{1}{2} \times c \times c = \frac{c^2}{2} \).

Thus, the total area where \( |x - y| \leq c \) is:
\[ \text{Area}_{|x - y| \leq c} = a^2 - \left( \frac{c^2}{2} + \frac{c^2}{2} \right) = a^2 - c^2 \]

### Step 7: Calculate the Area Where \( |x - y| > c \)
The area where \( |x - y| > c \) is:
\[ \text{Area}_{|x - y| > c} = a^2 - (a^2 - c^2) = c^2 \]

### Step 8: Calculate the Probability
The probability \( P \) that the distance between the two points exceeds \( c \) is given by the ratio of the area where \( |x - y| > c \) to the total area:
\[ P = \frac{\text{Area}_{|x - y| > c}}{\text{Total Area}} = \frac{c^2}{a^2} \]

### Final Answer
Thus, the probability that the distance between the two points exceeds \( c \) is:
\[ P = 1 - \left( \frac{c}{a} \right)^2 \]