Problem Analysis:

• We are given the ratio of two non-zero real variables, $x$ and $y$, in terms of a parameter $a > 1$. The given ratio is $x: y = (a + 1) : (a − 1)$.
• The objective is to find the ratio $(x^2 – y^2) : (x^2 + y^2)$ in terms of $a$.

Deriving the Ratio

From the given ratio, we can write:

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

Squaring both sides, we get the ratio of the squares:

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

Let $R = \frac{x^2}{y^2}$. We need to find the ratio $\frac{x^2 - y^2}{x^2 + y^2}$. To simplify this expression, we can divide both the numerator and the denominator by $y^2$:

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

Substituting and Simplifying

Now, substitute the expression for $R$:

$ R - 1 = \frac{(a + 1)^2}{(a - 1)^2} - 1 = \frac{(a + 1)^2 - (a - 1)^2}{(a - 1)^2} $

Using the identity $(A+B)^2 - (A-B)^2 = 4AB$, where $A=a$ and $B=1$:

$ (a + 1)^2 - (a - 1)^2 = 4a(1) = 4a $

So, $R - 1 = \frac{4a}{(a - 1)^2}$.

Similarly, calculate $R + 1$:

$ R + 1 = \frac{(a + 1)^2}{(a - 1)^2} + 1 = \frac{(a + 1)^2 + (a - 1)^2}{(a - 1)^2} $

Using the identity $(A+B)^2 + (A-B)^2 = 2(A^2 + B^2)$, where $A=a$ and $B=1$:

$ (a + 1)^2 + (a - 1)^2 = 2(a^2 + 1^2) = 2(a^2 + 1) $

So, $R + 1 = \frac{2(a^2 + 1)}{(a - 1)^2}$.

Final Ratio Calculation

Now, compute the required ratio $\frac{R - 1}{R + 1}$:

$ \frac{R - 1}{R + 1} = \frac{\frac{4a}{(a - 1)^2}}{\frac{2(a^2 + 1)}{(a - 1)^2}} $

Cancel out the common term $\frac{1}{(a - 1)^2}$ (since $a \neq 1$):

$ \frac{R - 1}{R + 1} = \frac{4a}{2(a^2 + 1)} $

Simplify the expression:

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

Therefore, the ratio $(x^2 – y^2) : (x^2 + y^2)$ is $2a : (a^2 + 1)$.