A line passes through the point `P (5,6)` outside the circle `x^(2) + y ^(2) = 12 ` and meets the circle at A and B. The value of PA. PB is equal to

Options

A. 25
B. 36
C. 49
D. 61

Answer

To solve the problem, we need to find the value of \( PA \cdot PB \) where \( P(5, 6) \) is a point outside the circle defined by the equation \( x^2 + y^2 = 12 \).

### Step-by-Step Solution:

1. **Identify the Circle's Equation**:
The equation of the circle is given by:
\[ x^2 + y^2 = 12 \]
This can be rewritten in standard form as:
\[ x^2 + y^2 - 12 = 0 \]

2. **Determine the Coordinates of Point P**:
The coordinates of point \( P \) are \( (5, 6) \).

3. **Use the Length of Tangent Formula**:
The length of the tangent \( PC \) from point \( P \) to the circle can be calculated using the formula:
\[ PC = \sqrt{S_1} \]
where \( S_1 \) is obtained by substituting the coordinates of point \( P \) into the circle's equation.

4. **Calculate \( S_1 \)**:
Substitute \( x = 5 \) and \( y = 6 \) into the equation of the circle:
\[ S_1 = 5^2 + 6^2 - 12 \]
Calculate:
\[ S_1 = 25 + 36 - 12 = 49 \]

5. **Find the Length of Tangent \( PC \)**:
Now, calculate \( PC \):
\[ PC = \sqrt{49} = 7 \]

6. **Calculate \( PA \cdot PB \)**:
According to the tangent-secant theorem, we have:
\[ PA \cdot PB = PC^2 \]
Thus:
\[ PA \cdot PB = 7^2 = 49 \]

### Final Answer:
The value of \( PA \cdot PB \) is \( 49 \).

---