If, for non-zero real variables x and y, and real parameter a 1,

x : y = (a+1) : (a-1),

then, the ratio (x^2 - y^...

Question

If, for non-zero real variables x and y, and 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)

Accepted Answer

(x^2 - y^2) : (x^2 + y^2) = 2a : (a^2 + 1)  [Answer A] We are given a ratio x:y expressed in terms of a parameter a, and we need to find a new ratio involving x^2 and y^2. The key identities are the difference-of-squares formula (x+y)(x-y) = x^2 - y^2, and the sum of squares formula x^2 + y^2. By substituting x = a+1 and y = a-1 (taking the simplest proportional values), we compute both expressions.