JEE Math Practice Question

Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ externally and also touches the $$x$$-axis be $$\mathrm{L}$$. Then the area bounded by $$\mathrm{L}$$ and the line $$y=4$$ is:

1. A.$$ \frac{32 \sqrt{2}}{3} $$
2. B.$$ \frac{40 \sqrt{2}}{3} $$
3. C.$$\frac{64}{3}$$
4. D.$$ \frac{32}{3} $$