If the differential equation $\frac{d y}{d x}+\frac{x}{y}=\frac{a}{y}$ where $a$ is constant, represents a family of circles then the radius of the circle is ......

The differential equation $\frac{dy}{dx} + \frac{x}{y} = \frac{a}{y}$ can be transformed by multiplying both sides by $y$ to eliminate denominators, yielding $y\frac{dy}{dx} + x = a$.

Rearranging terms separates the variables as $y\,dy = (a - x)\,dx$.

Integrating both sides gives $\frac{y^2}{2} = ax - \frac{x^2}{2} + C_1$, where $C_1$ is the constant of integration.

Multiplying through by 2 produces $y^2 = 2ax - x^2 + 2C_1$.

Completing the square for the $x$-terms by adding $a^2$ to both sides results in $(x - a)^2 + y^2 = a^2 - 2C_1$.

This represents a circle centered at $(a, 0)$ with radius $R = \sqrt{a^2 - 2C_1}$.

Introducing a new constant $c = -C_1$ simplifies the expression to $R = \sqrt{a^2 + 2c}$.

The solution corresponds to option B.