The harmonic mean of two positive numbers a and b is 4 , their arithmeitc mean is A and the geometric mean is G . If 2 A + G 2 = 27 , a + b = α and a - b = β , then the value of α β is equal to

Step-by-step solution

Given, harmonic mean H = 4 We know that, G 2 = A H Since, 2 A + G 2 = 2 7 ⇒ 2 A + A H = 2 7 ⇒ 2 A + 4 A = 2 7 ⇒ A = 27 6 = 9 2 = a + b 2 ⇒ a + b = 9 G 2 = A H = 9 2 × 4 = 1 8 ⇒ a b = 1 8 a - b = a + b 2 - 4 a b = 81 - 4 × 18 = 3 ⇒ α = 9 ,   β = 3 ⇒ α β = 3