Answer is: option2

-1

Solution:

We are given the function:

\[ f(x) = \frac{x}{x - \frac{a}{x}} \]

and we need to determine the value of \( a \) given that \( f'(1) = \frac{1}{2} \).

Rewriting \( f(x) \):

\[ f(x) = \frac{x}{x^2 - a} = \frac{x^2}{x^2 - a} \] Using the quotient rule: \[ f'(x) = \frac{(2x)(x^2 - a) - (x^2)(2x)}{(x^2 - a)^2} \] \[ = \frac{2x(x^2 - a) - 2x^3}{(x^2 - a)^2} \] \[ = \frac{2x^3 - 2ax - 2x^3}{(x^2 - a)^2} \] \[ = \frac{-2ax}{(x^2 - a)^2} \] Substituting \( x = 1 \): \[ f'(1) = \frac{-2a(1)}{(1^2 - a)^2} = \frac{-2a}{(1 - a)^2} \]

Given that \( f'(1) = \frac{1}{2} \), we set up the equation:

\[ \frac{-2a}{(1 - a)^2} = \frac{1}{2} \] \[ -2a = \frac{(1 - a)^2}{2} \] \[ 4a = -(1 - a)^2 \] \[ a^2 + 2a + 1 = 0 \] \[ (a + 1)^2 = 0 \] \[ a + 1 = 0 \] \[ a = -1 \] The correct answer is: \[ \boxed{-1} \quad \]