The centres of the circles are (a, c) and (b, c) and their radical axis is y-axis. The radius of one of the circles is r. The radius of the other circle is

Options

A. `r^2-a^2+b^2`
B. `2(r^2-a^2+b^2)`
C. `sqrt(r^2-a^2+b^2)`
D. `2sqrt(r^2-a^2+b^2)`

Answer

To solve the problem, we need to find the radius of the second circle given the radius of the first circle and the centers of both circles. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Centers and Radii**:
- The centers of the circles are given as \( (a, c) \) and \( (b, c) \).
- The radius of the first circle is given as \( r \).
- Let the radius of the second circle be \( R \).

2. **Write the Equations of the Circles**:
- The equation of the first circle centered at \( (a, c) \) with radius \( r \) is:
\[ (x - a)^2 + (y - c)^2 = r^2 \]
- The equation of the second circle centered at \( (b, c) \) with radius \( R \) is:
\[ (x - b)^2 + (y - c)^2 = R^2 \]

3. **Use the Radical Axis Concept**:
- The radical axis of two circles is the locus of points that have equal power with respect to both circles. Since the radical axis is the y-axis, we set \( x = 0 \).

4. **Substitute \( x = 0 \) into the Circle Equations**:
- For the first circle:
\[ (0 - a)^2 + (y - c)^2 = r^2 \implies a^2 + (y - c)^2 = r^2 \]
- For the second circle:
\[ (0 - b)^2 + (y - c)^2 = R^2 \implies b^2 + (y - c)^2 = R^2 \]

5. **Set Up the Equation for the Radical Axis**:
- The radical axis condition gives us:
\[ a^2 + (y - c)^2 - r^2 = b^2 + (y - c)^2 - R^2 \]
- Canceling \( (y - c)^2 \) from both sides:
\[ a^2 - r^2 = b^2 - R^2 \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ R^2 = b^2 - a^2 + r^2 \]

7. **Taking the Square Root**:
- Thus, the radius \( R \) of the second circle is:
\[ R = \sqrt{r^2 + b^2 - a^2} \]

### Final Answer:
The radius of the second circle is:
\[ R = \sqrt{r^2 + b^2 - a^2} \]