4. Two circles, the radii of which are in the ratio 2:3, touch each other internally. A line is drawn through the center of the smaller circle, perpendicular to the line of centers, and tangents are drawn from the points of intersection of this line with the larger circle to the smaller circle. Find the angles between these tangents.
Answer: All angles are right angles.

Solution. By the Pythagorean theorem, $O_{2} B=$ $\sqrt{9-1}=2 \sqrt{2}$. From the right triangle $O_{2} B M$, we find $B M=2$. Therefore, triangle $O_{2} B M$ is isosceles with an angle of $45^{\circ}$. Similarly, the angle $O_{2} B N$ is $45^{\circ}$.

Thus, the tangents drawn from one point to the smaller circle are perpendicular to each other.

![](https://cdn.mathpix.com/cropped/2024_05_06_da8342610bd92e1fa782g-2.jpg?height=551&width=602&top_left_y=335&top_left_x=1249)

Calculations can be performed in other ways as well.

Criteria. In the case of full justification - a score of 7 points.