The circle with its center at Point O (0; 1) passes through the point M (-2; 4). Find the diameter of this circle.

Let’s connect the points O and M with a segment OM. It is clear that OM is the radius of a circle centered at point O and passing through point M.
It is known that the distance d between the points (х1; у1) and (х2; у2) on the coordinate plane хОу can be determined by the formula d = √ [(x2 – x1) 2 + (y2 – y1) 2].
We have: ОМ = √ [(- 2 – 0) 2 + (4 – 1) 2] = √ (4 + 9) = √ (13). So, we found the length of the radius of the circle.
It is clear that to determine the length of the diameter of a circle, you need to multiply the length of the radius by 2. Therefore, the length of the circle is 2 * √ (13) = 2√ (13).
Answer: 2√ (13).