If [x] is read as the greatest integer less than or equal to x, {x} is the least integer greater than or equal to x. Further f(x,y)=[x]+{y}
and g(x,y)=[x]-{y}
and P(x,y)=f(x,y)+g(x,y)
and Q(x,y)=f(x,y)-g(x,y)
If `x^2=16` and `y^2=25`, P (x,y).Q (x,y) is :

Options

A. 80
B. -80
C. 72
D. none of these

Answer

To solve the problem step-by-step, we need to evaluate the functions \( P(x, y) \) and \( Q(x, y) \) based on the definitions given and the values of \( x \) and \( y \).

### Step 1: Determine the values of \( x \) and \( y \)
Given that \( x^2 = 16 \) and \( y^2 = 25 \):
- The possible values for \( x \) are \( x = 4 \) or \( x = -4 \).
- The possible values for \( y \) are \( y = 5 \) or \( y = -5 \).

### Step 2: Calculate \( f(x, y) \) and \( g(x, y) \)
The functions are defined as:
- \( f(x, y) = [x] + \{y\} \)
- \( g(x, y) = [x] - \{y\} \)

Where:
- \( [x] \) is the greatest integer less than or equal to \( x \).
- \( \{y\} \) is the least integer greater than or equal to \( y \).

### Step 3: Evaluate \( f(x, y) \) and \( g(x, y) \) for each combination of \( x \) and \( y \)

1. **For \( x = 4 \) and \( y = 5 \)**:
- \( [4] = 4 \)
- \( \{5\} = 5 \)
- Therefore, \( f(4, 5) = 4 + 5 = 9 \)
- \( g(4, 5) = 4 - 5 = -1 \)

2. **For \( x = 4 \) and \( y = -5 \)**:
- \( [4] = 4 \)
- \( \{-5\} = -5 \)
- Therefore, \( f(4, -5) = 4 - 5 = -1 \)
- \( g(4, -5) = 4 + 5 = 9 \)

3. **For \( x = -4 \) and \( y = 5 \)**:
- \( [-4] = -4 \)
- \( \{5\} = 5 \)
- Therefore, \( f(-4, 5) = -4 + 5 = 1 \)
- \( g(-4, 5) = -4 - 5 = -9 \)

4. **For \( x = -4 \) and \( y = -5 \)**:
- \( [-4] = -4 \)
- \( \{-5\} = -5 \)
- Therefore, \( f(-4, -5) = -4 - 5 = -9 \)
- \( g(-4, -5) = -4 + 5 = 1 \)

### Step 4: Calculate \( P(x, y) \) and \( Q(x, y) \)
Now we can calculate \( P(x, y) \) and \( Q(x, y) \):
- \( P(x, y) = f(x, y) + g(x, y) \)
- \( Q(x, y) = f(x, y) - g(x, y) \)

1. **For \( x = 4 \) and \( y = 5 \)**:
- \( P(4, 5) = 9 - 1 = 8 \)
- \( Q(4, 5) = 9 + 1 = 10 \)

2. **For \( x = 4 \) and \( y = -5 \)**:
- \( P(4, -5) = -1 + 9 = 8 \)
- \( Q(4, -5) = -1 - 9 = -10 \)

3. **For \( x = -4 \) and \( y = 5 \)**:
- \( P(-4, 5) = 1 - 9 = -8 \)
- \( Q(-4, 5) = 1 + 9 = 10 \)

4. **For \( x = -4 \) and \( y = -5 \)**:
- \( P(-4, -5) = -9 + 1 = -8 \)
- \( Q(-4, -5) = -9 - 1 = -10 \)

### Step 5: Calculate \( P(x, y) \cdot Q(x, y) \)
Now we calculate \( P(x, y) \cdot Q(x, y) \) for each combination:

1. **For \( x = 4 \) and \( y = 5 \)**:
- \( P(4, 5) \cdot Q(4, 5) = 8 \cdot 10 = 80 \)

2. **For \( x = 4 \) and \( y = -5 \)**:
- \( P(4, -5) \cdot Q(4, -5) = 8 \cdot (-10) = -80 \)

3. **For \( x = -4 \) and \( y = 5 \)**:
- \( P(-4, 5) \cdot Q(-4, 5) = -8 \cdot 10 = -80 \)

4. **For \( x = -4 \) and \( y = -5 \)**:
- \( P(-4, -5) \cdot Q(-4, -5) = -8 \cdot (-10) = 80 \)

### Conclusion
The possible values for \( P(x, y) \cdot Q(x, y) \) are \( 80 \) and \( -80 \).