If x is very small in magnitude compared with a
such that `((a)/(a+ x))^(1//2) + ((a)/(a-x))^(1//2) = 2 + k (x^(2))/(a^(2))`, then the value of k, is

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(3)/(4)`
D. 1

Answer

c

Explanation

We have,
`((a)/(a + x))^(1//2) + ((a )/(a-x) )^(1//2) = (1)/((1 + (x)/a)^(1//2))+(1)/((1 - (x)/(a))^(1//2))`
`rArr ((a)/(a + x))^(1//2) + ((a )/(a-x) )^(1//2) = (1 + (x)/(a))^(-1//2) + (1 - (x)/(a) )^(1//2)`
`rArr ((a)/(a + x))^(1//2) + ((a )/(a-x) )^(1//2) = (1 - (1)/(2) (x)/(a) + (3)/(8)(x^(2))/(a^(2))) + (1 + (1)/(2)(x)/(a) + (3)/(8) (x^(2))/(a^(2)))`
`rArr ((a )/(a + x))^(1//2) + ((a)/(a - x))^(1//2) = 2 + (3)/(4) (x^(2))/(a^(2))`
`therefore k = (3)/(4)`