If $a = \frac{x}{x+y}$ and $b = \frac{y}{x-y}$, then $\frac{ab}{a+b}$ is equal to

### Concept & Formula This problem tests the algebraic manipulation of complex fractions. To solve it efficiently, you must evaluate the numerator ($a \cdot b$) and the denominator ($a + b$) of the target expression separately, and then divide them. It heavily relies on recognizing the difference of squares identity. $$ (x-y)(x+y) = x^2 - y^2 $$ ### Step-by-Step Solution * **Given:** * $a = \frac{x}{x+y}$ * $b = \frac{y}{x-y}$ * Target expression: $\frac{ab}{a+b}$ * **Calculation:** 1. **Calculate the numerator ($ab$):** Multiply the two fractions together. $ab = \left(\frac{x}{x+y}\right) \cdot \left(\frac{y}{x-y}\right) = \frac{xy}{(x+y)(x-y)}$. Apply the difference of squares identity: $\frac{xy}{x^2 - y^2}$. 2. **Calculate the denominator ($a+b$):** Add the two fractions. $a+b = \frac{x}{x+y} + \frac{y}{x-y}$. Find a common denominator, which is $(x+y)(x-y)$. Multiply the first numerator by $(x-y)$ and the second by $(x+y)$: $\frac{x(x-y) + y(x+y)}{(x+y)(x-y)}$. Expand the numerator: $\frac{x^2 - xy + xy + y^2}{x^2 - y^2}$. Cancel the $-xy$ and $+xy$ terms: $\frac{x^2 + y^2}{x^2 - y^2}$. 3. **Combine the results:** Divide the calculated numerator by the calculated denominator. $\frac{ab}{a+b} = \frac{\frac{xy}{x^2 - y^2}}{\frac{x^2 + y^2}{x^2 - y^2}}$. 4. When dividing fractions with identical denominators, the denominators cancel out. The result is simply the top numerator over the bottom numerator: $\frac{xy}{x^2 + y^2}$. ### Exam Strategy & Shortcut For complex algebraic variable problems, use the Value Assumption method to bypass algebra entirely. Let $x = 2$ and $y = 1$. $a = \frac{2}{2+1} = \frac{2}{3}$. $b = \frac{1}{2-1} = 1$. Evaluate target: $\frac{(2/3)(1)}{(2/3) + 1} = \frac{2/3}{5/3} = \frac{2}{5}$. Now test the options with $x=2, y=1$: (a) $\frac{(2)(1)}{2^2 + 1^2} = \frac{2}{4+1} = \frac{2}{5}$. (Matches perfectly!) (b) $\frac{5}{2}$. (Incorrect) This method confirms the answer in under a minute with low risk of algebraic errors. ### Common Pitfall When finding the common denominator for $a+b$, students frequently forget to distribute the variables across the entire binomials (e.g., writing $x(x-y)$ as $x^2 - y$ instead of $x^2 - xy$). This completely breaks the simplification process. Always use parentheses when multiplying terms. ### Final Answer Therefore, the correct answer is $\frac{xy}{x^2 + y^2}$.