l,b are the length and breadth of a rectangle respectively . If the perimeter of this rectangle is numerically equal to the area of the rectangle . What is the value of l-b ( where `l gt b`) ?

Options

A. a.1
B. b.2
C. c.3
D. d.can't be determined

Answer

To solve the problem, we need to find the value of \( l - b \) given that the perimeter of a rectangle is numerically equal to its area. Here are the steps to derive the solution:

### Step 1: Write the formulas for perimeter and area
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) \]
The area \( A \) of a rectangle is given by:
\[ A = l \times b \]

### Step 2: Set the perimeter equal to the area
According to the problem, the perimeter is numerically equal to the area:
\[ 2(l + b) = l \times b \]

### Step 3: Rearrange the equation
We can rearrange the equation to isolate terms involving \( l \) and \( b \):
\[ l \times b - 2l - 2b = 0 \]

### Step 4: Rearrange into a standard quadratic form
Rearranging gives us:
\[ lb - 2l - 2b = 0 \]
This can be rewritten as:
\[ lb - 2l - 2b = 0 \]

### Step 5: Factor the equation
We can factor this equation:
\[ l(b - 2) - 2b = 0 \]
This implies:
\[ l(b - 2) = 2b \]
From this, we can express \( l \):
\[ l = \frac{2b}{b - 2} \]

### Step 6: Substitute \( l \) into \( l - b \)
Now we need to find \( l - b \):
\[ l - b = \frac{2b}{b - 2} - b \]
To combine the terms, we need a common denominator:
\[ l - b = \frac{2b - b(b - 2)}{b - 2} \]
This simplifies to:
\[ l - b = \frac{2b - b^2 + 2b}{b - 2} = \frac{4b - b^2}{b - 2} \]

### Step 7: Factor \( 4b - b^2 \)
Factoring out \( b \):
\[ l - b = \frac{b(4 - b)}{b - 2} \]

### Step 8: Analyze the expression
Since \( l > b \), we need \( 4 - b > 0 \) which implies \( b < 4 \). Thus, we can find specific values for \( b \) that satisfy \( l > b \).

### Conclusion
We can conclude that the value of \( l - b \) depends on the specific values of \( b \) chosen under the condition \( b < 4 \).