Two circles of radii 15 cm and 10 cm intersecr each and the length of their common chord is 16 cm. What is the distance (in cm) between their centres ?

Options

A. `6+sqrt(161)`
B. `15+2sqrt(161)`
C. `12+3sqrt(7)`
D. `10+sqrt(161)`

Answer

To solve the problem of finding the distance between the centers of two intersecting circles with given radii and the length of their common chord, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the first circle (R1) = 15 cm
- Radius of the second circle (R2) = 10 cm
- Length of the common chord (PQ) = 16 cm

2. **Find the Half-Length of the Common Chord:**
- Since the common chord is bisected by the line joining the centers of the two circles, the half-length (MN) of the common chord is:
\[ MN = \frac{PQ}{2} = \frac{16}{2} = 8 \text{ cm} \]

3. **Apply the Pythagorean Theorem in Triangle OMP:**
- In triangle OMP (where O is the center of the first circle, M is the midpoint of the chord, and P is a point on the circumference of the first circle), we can apply the Pythagorean theorem:
\[ OP^2 = OM^2 + MP^2 \]
- Here, \( OP = R1 = 15 \) cm, \( MP = MN = 8 \) cm, and \( OM \) is the distance from the center of the first circle to the midpoint of the chord.
- Substituting the known values:
\[ 15^2 = OM^2 + 8^2 \]
\[ 225 = OM^2 + 64 \]
\[ OM^2 = 225 - 64 = 161 \]
\[ OM = \sqrt{161} \text{ cm} \]

4. **Apply the Pythagorean Theorem in Triangle O'NQ:**
- Similarly, for the second circle, we can apply the Pythagorean theorem in triangle O'NQ (where O' is the center of the second circle, N is the midpoint of the chord, and Q is a point on the circumference of the second circle):
\[ O'N^2 = O'M^2 + MN^2 \]
- Here, \( O'N = R2 = 10 \) cm, \( MN = 8 \) cm, and \( O'M \) is the distance from the center of the second circle to the midpoint of the chord.
- Substituting the known values:
\[ 10^2 = O'M^2 + 8^2 \]
\[ 100 = O'M^2 + 64 \]
\[ O'M^2 = 100 - 64 = 36 \]
\[ O'M = \sqrt{36} = 6 \text{ cm} \]

5. **Calculate the Distance Between the Centers:**
- The distance (d) between the centers O and O' of the two circles can be found by adding the distances OM and O'M:
\[ d = OM + O'M = \sqrt{161} + 6 \text{ cm} \]

### Final Answer:
The distance between the centers of the two circles is:
\[ d = 6 + \sqrt{161} \text{ cm} \]