If $a$ is sufficiently large as compared with $b,$ and $\sqrt \frac{a}{a-b}+\sqrt \frac{a}{a+b}=2+k(\frac{b}{a})^2$,then what is the value of $k?$ 
If  $a$ is sufficiently large as compared with $b,$ and $\sqrt
 \frac{a}{a-b}+\sqrt \frac{a}{a+b}=2+k(\frac{b}{a})^2$,then what is the
value of $k?$

Query
Can we find it using particular value?,like putting $a=4,b=2,$ then we get

$k=4\sqrt2+4\sqrt \frac{2}{3}-8$

This problem is given in  Multiple Choice Question exercise,the possible answers given are
A.$\frac{2}{3}$
B.$\frac{3}{4}$
C.$\frac{4}{5}$
D.$\frac{5}{6}$
None of the answer mathches with mine....
Please give some hint to determine value of $k?$

Suppose that for small $\epsilon$:
$$(1+\epsilon)^{-1/2}=1+\binom{-1/2}{1}\epsilon+\binom{-1/2}{2}{\epsilon^2}+O(\epsilon^3)=1-\frac12\epsilon+\frac{3}{8}\epsilon^2+O(\epsilon^3)$$
If you put $-\epsilon$ instead of $\epsilon$ into the last expression you get:
$$(1-\epsilon)^{-1/2}=1+\frac12\epsilon+\frac{3}{8}e^2+O(\epsilon^3)$$
It means that:
$$(1+\epsilon)^{-1/2}+(1-\epsilon)^{-1/2}=2+\frac34\epsilon^2+O(\epsilon^3)$$
If you put $\epsilon=b/a$, you are done. Obviously $k=\frac34$
