Final answer:
The point (1,2) lies on a circle centered at the origin through the point (0,3).
Step-by-step explanation:
To determine if the point (1,2) lies on a circle centered at the origin through the point (0,3), we can calculate the distance between these two points and compare it to the radius of the circle. The distance between two points can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates, we have:
d = sqrt((1 - 0)^2 + (2 - 3)^2) = sqrt(1 + 1) = sqrt(2).
Therefore, the distance between the two points is sqrt(2). If the circle has its center at the origin, the radius of the circle is also sqrt(2) since the distance from the origin to any point on the circle is the same. Hence, the point (1,2) does lie on a circle centered at the origin through the point (0,3).