The third proportional to $(x^2 - y^2)$ and $(x - y)$ is

### Concept & Formula The third proportional $c$ to two given terms $a$ and $b$ is found using the continuous proportion formula $a : b :: b : c$, which means $c = \frac{b^2}{a}$. ### Step-by-Step Solution 1. Let the first term be $a = (x^2 - y^2)$ and the second term be $b = (x - y)$. 2. Let the third proportional be $c$. Set up the equation: $$ c = \frac{(x - y)^2}{(x^2 - y^2)} $$ 3. Factorize the denominator using the difference of squares identity, $x^2 - y^2 = (x - y)(x + y)$: $$ c = \frac{(x - y)(x - y)}{(x - y)(x + y)} $$ 4. Cancel the common term $(x - y)$ from the numerator and denominator: $$ c = \frac{x - y}{x + y} $$ ### Exam Strategy & Shortcut If algebra slows you down, plug in simple numbers (e.g., $x=3, y=1$). First term $= 3^2 - 1^2 = 8$. Second term $= 3 - 1 = 2$. Third proportional to 8 and 2 is $2^2 / 8 = 4/8 = 1/2$. Test options with $x=3, y=1$. Only $\frac{3-1}{3+1} = \frac{2}{4} = \frac{1}{2}$ matches the result. ### Common Pitfall Failing to factorize $x^2 - y^2$ correctly, or accidentally expanding $(x - y)^2$ in the numerator, which creates complex polynomials and makes simplification much harder. ### Final Answer Therefore, the correct answer is **$\frac{x - y}{x + y}$**.