The correct relation between $$\alpha$$ (ratio of collector current to emitter current) and $$\beta$$ (ratio of collector current to base current) of a transistor is :

In a transistor, the emitter current equals the sum of the collector current and the base current: $$I_E = I_C + I_B$$.

The current gain $$\alpha$$ is defined as $$\alpha = \frac{I_C}{I_E}$$ and $$\beta$$ is defined as $$\beta = \frac{I_C}{I_B}$$.

From $$I_E = I_C + I_B$$, dividing both sides by $$I_C$$: $$\frac{I_E}{I_C} = 1 + \frac{I_B}{I_C}$$, which gives $$\frac{1}{\alpha} = 1 + \frac{1}{\beta} = \frac{\beta + 1}{\beta}$$.

Solving for $$\alpha$$: $$\alpha = \frac{\beta}{1 + \beta}$$.