If `a gt b gt 0` are two real numbers, the value of, `sqrt(ab+(a-b)sqrt(ab+(a-b)sqrt(ab+(a-b)sqrt(ab+….))))` is

Options

A. independent of b
B. independent of a
C. independent of both a & b
D. dependent of both a & b

Explanation

To solve the expression \( y = \sqrt{ab + (a-b) \sqrt{ab + (a-b) \sqrt{ab + (a-b) \sqrt{ab + \ldots}}}} \), we can follow these steps:

### Step 1: Define the Expression
Let \( y \) be the value of the expression:
\[ y = \sqrt{ab + (a-b) \sqrt{ab + (a-b) \sqrt{ab + (a-b) \sqrt{ab + \ldots}}}} \]
This means that the expression inside the square root is also \( y \).

### Step 2: Rewrite the Expression
We can rewrite the expression as:
\[ y = \sqrt{ab + (a-b) y} \]

### Step 3: Square Both Sides
Next, we square both sides to eliminate the square root:
\[ y^2 = ab + (a-b) y \]

### Step 4: Rearrange the Equation
Rearranging gives us a quadratic equation:
\[ y^2 - (a-b)y - ab = 0 \]

### Step 5: Use the Quadratic Formula
We can solve this quadratic equation using the quadratic formula:
\[ y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \( A = 1 \), \( B = -(a-b) \), and \( C = -ab \). Plugging in these values:
\[ y = \frac{a-b \pm \sqrt{(a-b)^2 - 4(1)(-ab)}}{2(1)} \]
\[ y = \frac{a-b \pm \sqrt{(a-b)^2 + 4ab}}{2} \]

### Step 6: Simplify the Discriminant
Now, we simplify the discriminant:
\[ (a-b)^2 + 4ab = a^2 - 2ab + b^2 + 4ab = a^2 + 2ab + b^2 = (a+b)^2 \]
Thus, we have:
\[ y = \frac{a-b \pm (a+b)}{2} \]

### Step 7: Calculate the Possible Values
This gives us two possible solutions:
1. \( y = \frac{(a-b) + (a+b)}{2} = \frac{2a}{2} = a \)
2. \( y = \frac{(a-b) - (a+b)}{2} = \frac{-2b}{2} = -b \)

### Step 8: Determine the Valid Solution
Since \( y \) represents a square root, it must be non-negative. Therefore, we discard the negative solution:
\[ y = a \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{a} \]
This implies that the value of \( y \) is independent of \( b \).