The arithmetic mean of two positive numbers a and b exceeds their geometric mean by 3 2 and the geometric mean exceeds their harmonic mean by 6 5 . If a + b = α and a - b = β , then the value of 10 β α is equal to

Step-by-step solution

A - G = 3 2 and G - H = 6 5 Also, we know that G 2 = A H ⇒ G 2 = 3 2 + G G - 6 5 ⇒ G 2 = G 2 - 6 5 G + 3 2 G - 9 5 ⇒ 3 2 - 6 5 G = 9 5 ⇒ G = 9 5 × 10 3 = 6 G = 6 ⇒ A = 15 2 ⇒ a + b 2 = 15 2 and a b = 6 ⇒ a + b = 15 and a b = 36 a - b = a + b 2 - 4 a b = 225 - 4 × 36 = 9 α = a + b = 15 ,   β = a - b = 9 ⇒ 10 β α = 10 × 9 15 = 6