Question

answer the following ​

Answer 1

Let the mean proportion be z

(x+y)² : z :: z : (x-y)²

(x+y)²/z = z/(x-y)²

[cross multiplication]

(x+y)²(x-y)² = z×z

(x²+y²+2xy)(x²+y²-2xy) = z²

x²(x²+y²-2xy) +y²(x²+y²-2xy) +2xy(x²+y²-2xy) = z²

x⁴+x²y²-2x³y +x²y²+y⁴-2xy³ +2x³y+2xy³-4x²y² = z²

x⁴-2x²y²+y⁴ = z²

z = √(x⁴-2x²y²+y⁴)

z = √(x²-y²)²

z = (x²-y²)

Hence, Option (D) (x²-y²) is correct