If x, y, z are in continued proportion, then (y² + z²) : (x² + y²) is equal to :

1. z : x

2. x : z

3. x : y

4. (y + z) : (x + y)

Given,

x, y, z are in continued proportion

Let $\dfrac{x}{y} = \dfrac{y}{z} = k$ for some constant ratio k.

⇒ $\dfrac{x}{y} = k, \dfrac{y}{z} = k$

⇒ y = zk and x = yk = (zk)k = zk²

Substitute value of x and y in $\dfrac{y^2 + z^2}{x^2 + y^2}$ we get:

$\Rightarrow \dfrac{(zk)^2 + z^2}{(zk^2)^2 + (zk)^2} \\[1em] \Rightarrow \dfrac{z^2k^2 + z^2}{z^2k^4 + z^2k^2} \\[1em] \Rightarrow \dfrac{z^2(k^2 + 1)}{z^2k^2(k^2 + 1)} \\[1em] \Rightarrow \dfrac{1}{k^2} \\[1em] \Rightarrow \dfrac{1}{\dfrac{x}{y} \times \dfrac{y}{z}} \\[1em] \Rightarrow \dfrac{1}{\dfrac{x}{z}} \\[1em] \Rightarrow \dfrac{z}{x} \\[1em] \Rightarrow z : x.$

Hence, option 1 is the correct option.