Find the polar of the point (a, - b) with respect to the circle `x ^(2) + y^(2) + 2ax - 2by + a^(2) - b^(2) = 0`

Answer

To find the polar of the point (a, -b) with respect to the circle given by the equation \( x^2 + y^2 + 2ax - 2by + a^2 - b^2 = 0 \), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle Equation**:
The equation of the circle is given as:
\[ x^2 + y^2 + 2ax - 2by + a^2 - b^2 = 0 \]

2. **Define the Point**:
The point for which we need to find the polar is \( P(a, -b) \).

3. **Use the Polar Equation Formula**:
The polar of a point \( (x_1, y_1) \) with respect to a circle given by \( x^2 + y^2 + 2gx + 2fy + c = 0 \) is given by:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
Here, we need to identify \( g \), \( f \), and \( c \) from the circle equation.

4. **Extract Coefficients**:
From the circle equation, we can identify:
- \( g = a \)
- \( f = -b \)
- \( c = a^2 - b^2 \)

5. **Substitute into the Polar Equation**:
Substitute \( (x_1, y_1) = (a, -b) \) into the polar equation:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
becomes:
\[ x(a) + y(-b) + a(x + a) + (-b)(y - b) + (a^2 - b^2) = 0 \]

6. **Simplify the Equation**:
Expanding and simplifying:
\[ ax - by + ax + a^2 - by + b^2 + a^2 - b^2 = 0 \]
Combine like terms:
\[ 2ax - by + 2a^2 = 0 \]

7. **Final Polar Equation**:
Rearranging gives us the polar equation:
\[ ax - by + a^2 = 0 \]

### Final Answer:
The polar of the point \( (a, -b) \) with respect to the given circle is:
\[ ax - by + a^2 = 0 \]