Question

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

• A. $(x+y)$
• B. $x-y$
• C. $\frac{x+y}{x-y}$
• D. $\frac{x-y}{x+y}$

Answer 1

Answer: (D)

$\frac{x-y}{x+y}$

Let the third proportion to $(x^2-y^2)$ and $(x-y)$ be z. Then

$(x^2-y^2):(x-y)::(x-y):z$

or $\frac{x^2-y^2}{x-y}=\frac{x-y}{z}$

Cross-multiplying, we get

$(x^2-y^2)z=(x-y)(x-y)$

or $z=\frac{(x-y{)}^2}{x^2-y^2}=\frac{(x-y)}{(x+y)}$ as $x\ne y$

or $z=\frac{x-y}{x+y}$.