If $\frac{x}{y} = \frac{a + 2}{a - 2}$, then $\frac{x^2 - y^2}{x^2 + y^2}$ is equal to

### Concept & Logic This problem heavily relies on the mathematical property of proportions known as **Componendo and Dividendo**. $$\text{If } \frac{p}{q} = \frac{r}{s}, \text{ then } \frac{p + q}{p - q} = \frac{r + s}{r - s}$$ ### Step-by-Step Solution Given: $\frac{x}{y} = \frac{a + 2}{a - 2}$ First, square both sides to match the terms in the target expression ($x^2$ and $y^2$): $\frac{x^2}{y^2} = \frac{(a + 2)^2}{(a - 2)^2}$ Expand the terms on the right side using $(a \pm b)^2 = a^2 \pm 2ab + b^2$: $\frac{x^2}{y^2} = \frac{a^2 + 4a + 4}{a^2 - 4a + 4}$ Now, apply the rule of Componendo and Dividendo (in the subtract/add format: $\frac{\text{Num} - \text{Den}}{\text{Num} + \text{Den}}$): $\frac{x^2 - y^2}{x^2 + y^2} = \frac{(a^2 + 4a + 4) - (a^2 - 4a + 4)}{(a^2 + 4a + 4) + (a^2 - 4a + 4)}$ Simplify the numerator (subtracting the terms): $a^2 + 4a + 4 - a^2 + 4a - 4 = 8a$ Simplify the denominator (adding the terms): $a^2 + 4a + 4 + a^2 - 4a + 4 = 2a^2 + 8 = 2(a^2 + 4)$ Combine them back into the fraction: $\frac{8a}{2(a^2 + 4)}$ Divide numerator and denominator by 2 to simplify: $= \frac{4a}{a^2 + 4}$ ### Exam Strategy & Shortcut **Value Putting:** Assign a simple number to $a$. Let $a = 3$. Then $\frac{x}{y} = \frac{3 + 2}{3 - 2} = \frac{5}{1}$. So $x = 5$, $y = 1$. The required expression is $\frac{x^2 - y^2}{x^2 + y^2} = \frac{25 - 1}{25 + 1} = \frac{24}{26} = \frac{12}{13}$. Now, plug $a = 3$ into the options to see which one gives $\frac{12}{13}$: (a) $\frac{8(3)}{9 + 4} = \frac{24}{13}$ (Incorrect) (b) $\frac{4(3)}{9 - 4} = \frac{12}{5}$ (Incorrect) (d) $\frac{4(3)}{9 + 4} = \frac{12}{13}$ (Correct!) ### Common Pitfall Forgetting to square the right-hand side expressions before applying Componendo and Dividendo. Applying the rule immediately to $x/y$ instead of $x^2/y^2$ will result in completely wrong algebraic terms. ### Final Answer Therefore, the correct answer is **$\frac{4a}{a^2 + 4}$**.