Two circle touch each other externally at the point (0,k) and y-axis is the common tangent to these circles. Centres of these circle lie on the line

Options

A. `x =k`
B. `y=k`
C. `x + y=k`
D. `x -y=k`

Answer

To solve the problem, we need to determine the line on which the centers of two circles that touch each other externally at the point (0, k) lie, given that the y-axis is the common tangent to these circles.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two circles that touch each other externally at the point (0, k).
- The y-axis is a common tangent to both circles. This implies that the centers of both circles must be located at some distance away from the y-axis.

2. **Position of the Centers**:
- Let the centers of the two circles be \( C_1 \) and \( C_2 \).
- Since the y-axis is a common tangent, the centers \( C_1 \) and \( C_2 \) must be located horizontally away from the y-axis. This means that the x-coordinates of both centers must be positive or negative but not zero.

3. **Perpendicularity to the Tangent**:
- The radius drawn from the center of a circle to the point of tangency is perpendicular to the tangent line. Therefore, the line connecting the centers \( C_1 \) and \( C_2 \) must be horizontal (parallel to the x-axis) because the tangent (y-axis) is vertical.

4. **Determining the y-coordinate**:
- Since both circles touch at the point (0, k), and the line connecting the centers is horizontal, both centers must have the same y-coordinate, which is \( k \).

5. **Equation of the Line**:
- The line on which both centers lie can be expressed as \( y = k \). This indicates that regardless of the x-coordinates of \( C_1 \) and \( C_2 \), their y-coordinates are the same.

### Conclusion:
The centers of the circles lie on the line given by the equation \( y = k \).