Correct option(c) (5√6)/12
The arithmetic mean of two numbers a and b is
a + b/2
The geometric mean of two numbers a and b is
√ab
It is given that the arithmetic mean of two numbers a and b is five times their geometric mean. That is,
a + b/2 = 5√ab
⇒ a + b = 10√ab ....(1)
Squaring on both sides, of Eq. (1), we get
(a + b)2 = 100ab
⇒ a2 + b2 + 2ab = 100ab
⇒ a2 + b2 = 98ab
Subtracting -2ab from both sides of this equation, we get
a2 + b2 - 2ab = 98ab - 2ab
⇒ (a - b) 2 = 96ab
⇒ (a - b) = √96√ab ......(2)
Now,
Multiplying and dividing RHS by √6, we get