P is a point outside a circle and is 13 cm away from its centre. A secant drawn from the point P intersects the circle at points A and B in such a way that PA=9 cm and AB=7 cm. The radius of the circle is

Options

A. 5.5 cm
B. 5 cm
C. 4 cm
D. 4.5 cm

Answer

To find the radius of the circle given the conditions of the problem, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Distance from point P to the center O of the circle, \( OP = 13 \) cm.
- Length from P to A, \( PA = 9 \) cm.
- Length of segment AB, \( AB = 7 \) cm.

2. **Calculate PB:**
- Since \( PB = PA + AB \), we can calculate:
\[ PB = PA + AB = 9 \, \text{cm} + 7 \, \text{cm} = 16 \, \text{cm}. \]

3. **Use the Intersecting Secant Theorem:**
- According to the theorem, we have:
\[ PK \times PQ = PA \times PB. \]
- Here, \( PK = OP - OK \) and \( PQ = OP + OK \), where \( OK \) is the radius \( R \) of the circle.

4. **Express PK and PQ in terms of R:**
- Thus, we can express:
\[ PK = 13 - R, \]
\[ PQ = 13 + R. \]

5. **Substitute the Values into the Theorem:**
- Now substituting the values into the theorem:
\[ (13 - R)(13 + R) = 9 \times 16. \]

6. **Simplify the Left Side:**
- The left side can be simplified using the difference of squares:
\[ 13^2 - R^2 = 144. \]
- Calculating \( 13^2 \):
\[ 169 - R^2 = 144. \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ R^2 = 169 - 144, \]
\[ R^2 = 25. \]

8. **Finding the Radius R:**
- Taking the square root of both sides:
\[ R = \sqrt{25} = 5 \, \text{cm}. \]

### Final Answer:
The radius of the circle is \( R = 5 \, \text{cm} \).

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