In a binomial expansion of $$\left( x + y \right)^{2 n + 1} \left( x - y \right)^{2 n + 1}$$,the sum of middle terms is zero. What is the value of $$\left( {x^2 \over y^2} \right)$$

Correct option is A

Given: 

​$$\left( x + y \right)^{2n+1} \left( x^2 - y^2 \right)^{2x+1} \rightarrow \text{(odd)} \\T_{n+1} + T_{n+2} = 0\\\text{Solution:}\\\text{Let, } n = 1\\\left( x^2 - y^2 \right)^3\\\Rightarrow \left( x^2 \right)^3 - 3 \left( x^2 \right)^2 y^2 + 3 x^2 \left( y^2 \right)^2 - \left( y^2 \right)^3\\\Rightarrow -3x^4y^2 + 3x^2y^4 = 0\\3x^4y^2 = 3x^2y^4\\x^2 = y^2\\\frac{x^2}{y^2} = 1$$​​