The image you uploaded contains a mathematical expression and question in Turkish. Let's break it down:

We have: $a + b = a^2 - b^2$ It is given that $a$ and $b$ are positive integers.

Then, the problem asks for the value of the ratio: $\frac{a}{b} = \frac{ab}{a + b}$ The question is asking for the value of the ratio $\frac{a}{b}$ in terms of options:

• A) -1
• B) -1/2
• C) 1/2
• D) 1
• E) 1/2

We will first work through the given equation to find $\frac{a}{b}$.

Step 1: Simplify the equation $a + b = a^2 - b^2$

The right side of the equation is a difference of squares, so we can factor it: $a + b = (a - b)(a + b)$

Now, cancel $a + b$ from both sides (we know $a + b \neq 0$ because both $a$ and $b$ are positive): $1 = a - b$ This means: $a = b + 1$

Step 2: Substitute $a = b + 1$ into the ratio

Now, substitute $a = b + 1$ into the ratio $\frac{a}{b}$: $\frac{a}{b} = \frac{b + 1}{b} = 1 + \frac{1}{b}$

This is the value of $\frac{a}{b}$. Since there are no specific values for $b$, we need to check each option against the possible outcomes. However, the ratio will depend on the value of $b$, and the correct answer must fit the given choices.

Step 3: Check the value of $b$

Let's examine the equation for small values of $b$:

• If $b = 1$, then $a = 2$, and the ratio $\frac{a}{b} = \frac{2}{1} = 2$, which is not an option.
• If $b = 2$, then $a = 3$, and the ratio $\frac{a}{b} = \frac{3}{2}$, which is not an option.

Let's continue with more testing or see if any other method might be helpful!

Would you like me to proceed further, or would you like a deeper explanation on the steps?