If (x + y) : (x − y) = 4 : 1, then (x² + y²) : (x² − y²) is :

1. 8 : 17

2. 17 : 8

3. 16 : 1

4. 25 : 9

Given,

⇒ (x + y) : (x − y) = 4 : 1

$\Rightarrow \dfrac{(x + y)}{(x - y)} = \dfrac{4}{1}$

Applying Componendo and Dividendo, we get :

$\Rightarrow \dfrac{(x + y) + (x - y)}{(x + y) - (x - y)} = \dfrac{4 + 1}{4 - 1} \\[1em] \Rightarrow \dfrac{x + y + x - y}{x + y - x + y} = \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{2x}{2y} = \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{5}{3}.$

Let x = 5k and y = 3k for some constant k.

Substituting value of x and y in $\dfrac{x^2 + y^2}{x^2 - y^2}$, we get:

$\Rightarrow \dfrac{(5k)^2 + (3k)^2}{(5k)^2 - (3k)^2} \\[1em] \Rightarrow \dfrac{25k^2 + 9k^2}{25k^2 - 9k^2} \\[1em] \Rightarrow \dfrac{34k^2}{16k^2} \\[1em] \Rightarrow \dfrac{17}{8}.$

Thus, (x² + y²) : (x² − y²) = 17 : 8.

Hence, option 2 is the correct option.