Let $$y = y(x)$$ be a function of $$x$$ satisfying $$y\sqrt{1 - x^2} = k - x\sqrt{1 - y^2}$$ where $$k$$ is a constant and $$y\left(\frac{1}{2}\right) = -\frac{1}{4}$$. Then $$\frac{dy}{dx}$$ at $$x = \frac{1}{2}$$, is equal to

A

$$-\frac{\sqrt{5}}{4}$$

B

$$-\frac{\sqrt{5}}{2}$$

C

$$\frac{2}{\sqrt{5}}$$

D

$$\frac{\sqrt{5}}{2}$$