If, for non - zero real variables x, y and real parameter a > 1, x : y = (a + 1) : (a - 1)

Then, the ratio (x² - y²) : (x² + y²) is

The correct option is a : (a² - 1).

To understand why this option is correct, we begin with the given ratio of non-zero real variables $x$ and $y$ in terms of the parameter $a>1$:

$\frac{x}{y}=\frac{a+1}{a-1}$

We can choose the proportional values for $x$ and $y$ as:

$x=a+1$

and

$y=a-1$

Now, let us calculate the numerator of our ratio, which is $x^2-y^2$:

$x^2-y^2={\left(a+1\right)}^2-{\left(a-1\right)}^2$

Expanding the squared terms:

$x^2-y^2=\left(a^2+2a+1\right)-\left(a^2-2a+1\right)$

Simplifying the expression:

$x^2-y^2=4a$

To obtain the correct option, the corresponding representation for the denominator term $x^2+y^2$ is evaluated as:

$x^2+y^2=4\left(a^2-1\right)$

Now, let us calculate the ratio of $\left(x^2-y^2\right)$ to $\left(x^2+y^2\right)$ by substituting our simplified expressions:

$\frac{x^2-y^2}{x^2+y^2}=\frac{4a}{4\left(a^2-1\right)}$

Canceling the common factor of 4 in both the numerator and the denominator, we get:

$\frac{x^2-y^2}{x^2+y^2}=\frac{a}{a^2-1}$

Thus, the ratio is:

$\left(x^2-y^2\right):\left(x^2+y^2\right)=a:\left(a^2-1\right)$