If ax - by = a² + b² and $\dfrac{x + y}{2} = a$, then the value of x - y is :

1. a

2. 2a

3. b

4. 2b

Given,

Equations: ax - by = a² + b² and $\dfrac{x + y}{2} = a$

Solving equation $\dfrac{x + y}{2} = a$,

$\Rightarrow \dfrac{x + y}{2} = a \\[1em] \Rightarrow x + y = a \times 2 \\[1em] \Rightarrow x + y = 2a \\[1em] \Rightarrow y = 2a - x \text{ ……(1) }$

⇒ ax - by = a² + b²     …….(2)

Substituting value of y from equation (1) in (2) we get,

⇒ ax - by = a² + b²

⇒ ax - b(2a - x) = a² + b²

⇒ ax - 2ab + bx = a² + b²

⇒ ax + bx = a² + b² + 2ab

⇒ x(a + b) = a² + b² + 2ab

⇒ x(a + b) = (a + b)²

⇒ x = $\dfrac{(a + b)^2}{a + b}$

⇒ x = (a + b).

Substituting value of x in equation (1),

⇒ y = 2a - x

⇒ y = 2a - (a + b)

⇒ y = 2a - a - b

⇒ y = a - b

Now,

⇒ x - y = (a + b) - (a - b)

⇒ x - y = (a + b - a + b)

⇒ x - y = 2b.

Hence, option 4 is the correct option.