If $\frac{a}{b} + \frac{b}{a} = 2$, then the value of $(a - b)$ is

### Concept & Formula This problem elegantly masks a core **Algebraic Identity**: $$ (a - b)^2 = a^2 - 2ab + b^2 $$ By finding a common denominator for the given expression, it transforms directly into a perfect square trinomial, revealing the exact relationship between the variables $a$ and $b$. ### Step-by-Step Solution **Given:** $$ \frac{a}{b} + \frac{b}{a} = 2 $$ **Calculation:** Step 1: Find a common denominator ($ab$) for the left side of the equation. $$ \frac{a \times a + b \times b}{ab} = 2 $$ $$ \frac{a^2 + b^2}{ab} = 2 $$ Step 2: Multiply both sides by $ab$. $$ a^2 + b^2 = 2ab $$ Step 3: Move all terms to one side to form a quadratic expression. $$ a^2 - 2ab + b^2 = 0 $$ Step 4: Recognize the perfect square identity. $$ (a - b)^2 = 0 $$ Step 5: Take the square root of both sides. $$ a - b = 0 $$ ### Exam Strategy & Shortcut **The Value Assumption Method:** In competitive exams, when you see symmetrical algebraic equations that equal 2, test the assumption that the variables are equal to 1. Let $a = 1$ and $b = 1$. Check the condition: $\frac{1}{1} + \frac{1}{1} = 1 + 1 = 2$. It satisfies the given condition perfectly! Now, simply plug these values into the target expression: $(a - b) = 1 - 1 = 0$. This takes less than 3 seconds. ### Common Pitfall Students often try to isolate one variable in terms of the other (like $a = 2b - \frac{b^2}{a}$) and end up in endless loops of substitution. Never isolate variables in symmetric fraction sums; immediately cross-multiply to look for standard identities. ### Final Answer **Therefore, the correct answer is 0.**