Question

If, for non-zero real variables $x, y$ and a real parameter $a>1$, $x : y = (a+1) : (a-1)$. Then, the ratio $(x^2 - y^2) : (x^2 + y^2)$ is:

• A. $2a : (a^2+1)$
• B. $a : (a^2+1)$
• C. $2a : (a^2-1)$
• D. $a : (a^2-1)$

Hint:

For ratios involving squares in fractions, use substitution and simplifications effectively to simplify the terms systematically.

Answer 1

The Correct Option is A

Step 1: From the given condition, $x : y = (a+1) : (a-1)$, we write: $$\frac{x}{y} = \frac{a+1}{a-1}.$$ Step 2: Substitute this into the ratio $(x^2 - y^2) : (x^2 + y^2)$: $$\frac{x^2 - y^2}{x^2 + y^2} = \frac{\left(\frac{x}{y}\right)^2 - 1}{\left(\frac{x}{y}\right)^2 + 1}.$$ Step 3: Replace $\frac{x}{y}$ with $\frac{a+1}{a-1}$: $$\frac{\left(\frac{a+1}{a-1}\right)^2 - 1}{\left(\frac{a+1}{a-1}\right)^2 + 1} = \frac{\frac{(a+1)^2}{(a-1)^2} - 1}{\frac{(a+1)^2}{(a-1)^2} + 1}.$$ Simplify: $$\frac{\frac{(a+1)^2 - (a-1)^2}{(a-1)^2}}{\frac{(a+1)^2 + (a-1)^2}{(a-1)^2}} = \frac{(a+1)^2 - (a-1)^2}{(a+1)^2 + (a-1)^2}.$$ Step 4: Expand and simplify: $$(a+1)^2 - (a-1)^2 = 4a, \quad (a+1)^2 + (a-1)^2 = 2a^2 + 2.$$ So the ratio becomes: $$\frac{4a}{2a^2 + 2} = \frac{2a}{a^2+1}.$$ Step 5: The required ratio is: $$2a : (a^2 + 1).$$