1.9. Two circles touch each other externally. A tangent is drawn to the first circle, passing through the center of the second circle. The distance from the point of tangency to the center of the second circle is three times the radius of this circle. How many times greater is the length of the first circle compared to the length of the second circle?
1.9. Let $O_{1}$ and $O_{2}$ be the centers of the circles, and $A$ be the point of tangency (Fig. 1.8). Then $O_{1} A = R_{1}, O_{1} O_{2} = R_{1} + R_{2}, O_{2} A = 3 R_{2}$ (by the condition). We need to find the ratio $2 \pi R_{1} : 2 \pi R_{2} = R_{1} : R_{2}$. In the right triangle $O_{1} A O_{2} \left(\angle A = 90^{\circ}\right)$, we have

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-418.jpg?height=242&width=354&top_left_y=621&top_left_x=798)

Fig. 1.8

$$
O_{1} O_{2}^{2} = O_{1} A^{2} + O_{2} A^{2}, \text{ or } \left(R_{1} + R_{2}\right)^{2} = R_{1}^{2} + \left(3 R_{2}\right)^{2}
$$

Simplifying this equation, we get $R_{1} = 4 R_{2}$, hence $R_{1} : R_{2} = 4$.

Answer: 4 times.