In the given figure (circle), `P T=5,P D=7a n dP A=2,` then the value of `P B-P C=?` Fig

Answer

To solve the problem, we will use the properties of tangents and secants in a circle. Here are the steps to find the value of \( P B - P C \):

### Step 1: Understand the given information
We are given:
- \( PT = 5 \)
- \( PD = 7 \)
- \( PA = 2 \)

### Step 2: Use the tangent-secant theorem
According to the tangent-secant theorem, we have:
\[ PT^2 = PA \cdot PB \]
and
\[ PT^2 = PC \cdot PD \]

### Step 3: Calculate \( PT^2 \)
First, we calculate \( PT^2 \):
\[ PT^2 = 5^2 = 25 \]

### Step 4: Set up the equations
Using the tangent-secant theorem, we can set up the following equations:
1. From \( PT^2 = PA \cdot PB \):
\[ 25 = 2 \cdot PB \quad \text{(1)} \]
2. From \( PT^2 = PC \cdot PD \):
\[ 25 = PC \cdot 7 \quad \text{(2)} \]

### Step 5: Solve for \( PB \) and \( PC \)
From equation (1):
\[ PB = \frac{25}{2} \]

From equation (2):
\[ PC = \frac{25}{7} \]

### Step 6: Calculate \( PB - PC \)
Now, we find \( PB - PC \):
\[ PB - PC = \frac{25}{2} - \frac{25}{7} \]

### Step 7: Find a common denominator
The common denominator for 2 and 7 is 14. Thus, we rewrite the fractions:
\[ PB - PC = \frac{25 \cdot 7}{14} - \frac{25 \cdot 2}{14} = \frac{175 - 50}{14} = \frac{125}{14} \]

### Final Answer
The value of \( PB - PC \) is:
\[ PB - PC = \frac{125}{14} \]