In a rectangle the ratio of the length is to breadth is same as that of the sum of the length and breadth to the length . If l and b be the length and breadth of the rectangle then which of the following is true ?
(i) `(l)/(b) = (l^2)/(b^(2)) + 1 ` (ii) ` (b)/(l-b) = (l+b)/(l)` (iii) lb=(l+b)(l-b)`

Options

A. a.only (i) is true
B. b.only (ii) is true
C. c.only (ii) and (iii) are true
D. d.only (i) and (ii) are true

Answer

To solve the problem, we start with the given condition about the rectangle. We need to analyze the statement that the ratio of the length (l) to the breadth (b) is the same as the ratio of the sum of the length and breadth to the length.

### Step 1: Set up the equation
From the problem, we have:
\[ \frac{l}{b} = \frac{l + b}{l} \]

### Step 2: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ l \cdot l = b \cdot (l + b) \]
This simplifies to:
\[ l^2 = b(l + b) \]

### Step 3: Expand the right side
Expanding the right side, we get:
\[ l^2 = bl + b^2 \]

### Step 4: Rearranging the equation
Rearranging this equation gives:
\[ l^2 - bl - b^2 = 0 \]

### Step 5: Factor the quadratic equation
This is a quadratic equation in terms of l. We can factor it as:
\[ (l - b)(l + b) = 0 \]
This gives us two possible solutions:
1. \( l = b \)
2. \( l = -b \) (not applicable since length cannot be negative)

### Step 6: Analyze the conditions
Now we check the conditions given in the options:

1. **Condition (i)**: \( \frac{l}{b} = \frac{l^2}{b^2} + 1 \)

Substituting \( l = b \):
\[ \frac{b}{b} = \frac{b^2}{b^2} + 1 \implies 1 = 1 + 1 \quad \text{(False)} \]

2. **Condition (ii)**: \( \frac{b}{l - b} = \frac{l + b}{l} \)

Substituting \( l = b \):
\[ \frac{b}{b - b} = \frac{b + b}{b} \quad \text{(Undefined on the left side)} \]

3. **Condition (iii)**: \( lb = (l + b)(l - b) \)

Substituting \( l = b \):
\[ b \cdot b = (b + b)(b - b) \implies b^2 = 0 \quad \text{(False)} \]

### Conclusion
From the analysis, we find that only Condition (ii) holds true under the given conditions.