The third proportional to (x² − y²) and (x − y) is :

1. $\dfrac{x - y}{x + y}$

2. $\dfrac{x + y}{x - y}$

3. (x + y)

4. (x − y)

Let third proportional be p.

⇒ (x² − y²) : (x − y) = (x − y) : p

$\Rightarrow \dfrac{x^2 - y^2}{x - y} = \dfrac{x - y}{p} \\[1em] \Rightarrow p = \dfrac{(x - y)^2}{x^2 - y^2} \\[1em] \Rightarrow p = \dfrac{(x - y)^2}{(x + y)(x - y)} \\[1em] \Rightarrow p = \dfrac{x - y}{x + y}.$

Hence, option 1 is the correct option.