If from any point P, tangents PT, PT' are drawn to two given circles with centres A and B respectively, and if PN is the perpendicular from P on their radical axis, then `PT^(2) – PT'^(2)` =

Options

A. PN. AB
B. 2PN. AB
C. 4PN. AB
D. None of these

Answer

To solve the problem, we need to find the value of \( PT^2 - PT'^2 \), where \( PT \) and \( PT' \) are the lengths of the tangents drawn from point \( P \) to two circles centered at points \( A \) and \( B \) respectively.

### Step-by-step Solution:

1. **Identify the Circles**:
Let the equations of the two circles be:
\[ S: x^2 + y^2 + 2g_1x + c_1 = 0 \]
\[ S': x^2 + y^2 + 2g_2x + c_2 = 0 \]
Here, the centers of the circles are \( A(-g_1, 0) \) and \( B(-g_2, 0) \).

2. **Find the Lengths of the Tangents**:
The length of the tangent from point \( P(x_1, y_1) \) to the first circle \( S \) is given by:
\[ PT = \sqrt{x_1^2 + y_1^2 + 2g_1x_1 + c_1} \]
The length of the tangent from point \( P \) to the second circle \( S' \) is:
\[ PT' = \sqrt{x_1^2 + y_1^2 + 2g_2x_1 + c_2} \]

3. **Square the Tangent Lengths**:
We need to calculate \( PT^2 \) and \( PT'^2 \):
\[ PT^2 = x_1^2 + y_1^2 + 2g_1x_1 + c_1 \]
\[ PT'^2 = x_1^2 + y_1^2 + 2g_2x_1 + c_2 \]

4. **Calculate \( PT^2 - PT'^2 \)**:
Subtract the two equations:
\[ PT^2 - PT'^2 = (x_1^2 + y_1^2 + 2g_1x_1 + c_1) - (x_1^2 + y_1^2 + 2g_2x_1 + c_2) \]
Simplifying this gives:
\[ PT^2 - PT'^2 = (2g_1 - 2g_2)x_1 + (c_1 - c_2) \]

5. **Use the Radical Axis**:
The radical axis of the two circles is given by the equation:
\[ S - S' = 0 \implies (2g_1 - 2g_2)x + (c_1 - c_2) = 0 \]
This shows that the radical axis is a line where the difference in power of point \( P \) with respect to both circles is constant.

6. **Final Expression**:
Thus, we can express \( PT^2 - PT'^2 \) as:
\[ PT^2 - PT'^2 = 2(g_1 - g_2)x_1 \]
where \( x_1 \) is the perpendicular distance from point \( P \) to the radical axis.

### Conclusion:
The final expression for \( PT^2 - PT'^2 \) is:
\[ PT^2 - PT'^2 = 2(g_1 - g_2) \cdot PN \]
where \( PN \) is the perpendicular distance from point \( P \) to the radical axis.