By using property $f(x)= f(a-x)$
$f(2-x) =\int \limits_0^2 \large\frac{\sqrt {2-2+x}}{\sqrt {2-2+x}+\sqrt {2-x}}$$dx$---(1)
$\qquad =\int \limits_0^2 \large\frac{\sqrt x}{\sqrt x+\sqrt {2-x}}$$dx$----(2)
adding (1) and (2)
$2f(x)=\int \limits_0^2 dx$
$f(x)=\bigg[\large\frac{x}{2}\bigg]_0^2$
$f(x) => 1$
Hence a is the correct answer.