If two equal circles are such that the centre of one lies on e circumference of the other, then the ratio of the common chord of two circles to the radius of any of the circles is :

Options

A. `sqrt3:2`
B. `sqrt3:1`
C. `sqrt5:1`
D. none of these

Answer

To solve the problem, we need to find the ratio of the length of the common chord of two equal circles to the radius of any of the circles, given that the center of one circle lies on the circumference of the other.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the radius of each circle be \( r \).
- Circle 1 has its center at point \( O \) and Circle 2 has its center at point \( O' \).
- Since the center of Circle 2 lies on the circumference of Circle 1, the distance \( OO' = r \).

2. **Drawing the Common Chord**:
- Draw the common chord \( AB \) of the two circles.
- The line segment \( OO' \) will bisect the common chord \( AB \) perpendicularly at point \( P \).

3. **Applying the Right Triangle**:
- In the right triangle \( OAP \):
- \( OA = r \) (radius of Circle 1)
- \( OP = \frac{AB}{2} \) (half of the common chord)
- \( AP \) can be calculated using the Pythagorean theorem:
\[ OA^2 = OP^2 + AP^2 \]
Substituting the known values:
\[ r^2 = OP^2 + AP^2 \]

4. **Finding \( OP \)**:
- Since \( O'P = OP \) and \( OO' = r \), we can find \( OP \) as follows:
- The distance \( OP \) can be derived from the triangle formed by \( O \), \( O' \), and \( P \).
- The distance \( OP \) can be expressed as:
\[ OP = \sqrt{r^2 - \left(\frac{r}{2}\right)^2} \]
Simplifying:
\[ OP = \sqrt{r^2 - \frac{r^2}{4}} = \sqrt{\frac{3r^2}{4}} = \frac{r\sqrt{3}}{2} \]

5. **Calculating the Length of the Common Chord \( AB \)**:
- Since \( AB = 2 \times AP \) and \( AP = OP \):
\[ AB = 2 \times \frac{r\sqrt{3}}{2} = r\sqrt{3} \]

6. **Finding the Ratio**:
- The ratio of the common chord \( AB \) to the radius \( r \) is:
\[ \text{Ratio} = \frac{AB}{r} = \frac{r\sqrt{3}}{r} = \sqrt{3} \]
- Therefore, the ratio of the common chord to the radius of the circle is:
\[ \text{Ratio} = \sqrt{3} : 1 \]

### Final Answer:
The ratio of the common chord of the two circles to the radius of any of the circles is \( \sqrt{3} : 1 \).