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: \[ 2a : (a^2 + 1). \]