If l, b, p be the length, breadth and perimeter of a rectangle and b, l, p are in GP (in order), then l/b is:

Options

A. 2:1
B. `(sqrt3-1):1`
C. `(sqrt3+1):1`
D. `2:sqrt3`

Answer

To solve the problem, we need to analyze the given information about the rectangle and the relationship between its dimensions.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Let \( l \) be the length of the rectangle.
- Let \( b \) be the breadth of the rectangle.
- The perimeter \( p \) of a rectangle is given by the formula:
\[ p = 2(l + b) \]

2. **Identify the Geometric Progression (GP)**:
- We are told that \( b, l, p \) are in geometric progression (GP).
- For three numbers to be in GP, the square of the middle term must be equal to the product of the other two terms:
\[ l^2 = b \cdot p \]

3. **Substitute the Perimeter**:
- Substitute the expression for \( p \) into the GP condition:
\[ l^2 = b \cdot (2(l + b)) \]
- This simplifies to:
\[ l^2 = 2b(l + b) \]

4. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ l^2 = 2bl + 2b^2 \]
- Rearranging further, we get:
\[ l^2 - 2bl - 2b^2 = 0 \]

5. **Using the Quadratic Formula**:
- This is a quadratic equation in terms of \( l \). We can solve for \( l \) using the quadratic formula:
\[ l = \frac{-(-2b) \pm \sqrt{(-2b)^2 - 4 \cdot 1 \cdot (-2b^2)}}{2 \cdot 1} \]
- Simplifying this gives:
\[ l = \frac{2b \pm \sqrt{4b^2 + 8b^2}}{2} \]
- Further simplifying:
\[ l = \frac{2b \pm \sqrt{12b^2}}{2} \]
\[ l = b \pm b\sqrt{3} \]

6. **Finding the Ratio \( \frac{l}{b} \)**:
- We can express \( \frac{l}{b} \):
\[ \frac{l}{b} = 1 \pm \sqrt{3} \]

### Final Result:
Thus, the value of \( \frac{l}{b} \) is \( 1 + \sqrt{3} \) or \( 1 - \sqrt{3} \).