## Task 1/69

For two non-intersecting circles of different sizes, the common tangents are drawn. The angle between the external tangents is $\alpha$, and the angle between the internal tangents is $\beta$.

The angle between the tangents from the center of the larger circle to the smaller circle is sought.
We use the designations according to the figure. Then we have

$$
P O - N O - N P = \frac{R}{\sin \frac{\alpha}{2}} - \frac{r}{\sin \frac{\alpha}{2}} = M O + M P = \frac{R}{\sin \frac{\beta}{2}} + \frac{r}{\sin \frac{\beta}{2}}
$$

From this, it follows that \( P O \sin \frac{\alpha}{2} = R - r \quad (I) \quad \) and \( P O \sin \frac{\beta}{2} = R + r \quad (II) \). From (I) and (II), we get \( 2 r = P O \left( \sin \frac{\beta}{2} - \sin \frac{\alpha}{2} \right) \) and further

$$
\sin \frac{x}{2} = \frac{r}{P O} = \frac{1}{2} \left( \sin \frac{\beta}{2} - \sin \frac{\alpha}{2} \right)
$$

Thus, for the desired angle \( x \):

$$
x = 2 \arcsin \left[ \frac{1}{2} \left( \sin \frac{\beta}{2} - \sin \frac{\alpha}{2} \right) \right]
$$