Introduction:
The given problem states that a point P moves in such a way that the ratio of its distances from two coplanar points is always a fixed number. We need to determine the locus of point P.

Understanding the problem:
To understand the problem better, let's consider two coplanar points A and B. The ratio of the distances of point P from A and B is constant. Let's assume this constant ratio to be k.

Construction:
To solve the problem, we need to construct a diagram. Consider two coplanar points A and B. Draw a line segment AB and label its midpoint as M. Now, draw a perpendicular bisector to AB passing through point M. This perpendicular bisector will intersect AB at point P.

Properties of the perpendicular bisector:
1. The perpendicular bisector of a line segment AB passes through the midpoint M of AB.
2. The perpendicular bisector is equidistant from points A and B.

Deriving the locus:
Let's analyze the given scenario using the properties of the perpendicular bisector.

1. As P moves along the perpendicular bisector, it is equidistant from points A and B. This means the ratio of its distances from A and B is always 1.

2. If P is not on the perpendicular bisector, then it will be closer to either A or B. This will result in a ratio of distances different from 1.

Conclusion:
Based on the above analysis, we can conclude that the locus of point P is the perpendicular bisector of line segment AB.