To solve the problem of finding the equation of the polar of given points with respect to specified circles, we can follow these steps:
### (i) For the point (3, -1) with respect to the circle \(2x^2 + 2y^2 = 11\):
1. **Identify the point and circle**: The point is \( (3, -1) \) and the circle's equation can be simplified to \( x^2 + y^2 = \frac{11}{2} \).
2. **Use the polar equation formula**: The polar of a point \( (x_1, y_1) \) with respect to a circle \( x^2 + y^2 = r^2 \) is given by:
\[
x \cdot x_1 + y \cdot y_1 = r^2
\]
Here, \( r^2 = \frac{11}{2} \).
3. **Substituting values**:
\[
x \cdot 3 + y \cdot (-1) = \frac{11}{2}
\]
This simplifies to:
\[
3x - y = \frac{11}{2}
\]
4. **Multiply through by 2 to eliminate the fraction**:
\[
6x - 2y = 11
\]
### Final Equation:
\[
6x - 2y - 11 = 0
\]
---
### (ii) For the point (2, 3) with respect to the circle \(x^2 + y^2 + 6x + 8y - 96 = 0\):
1. **Identify the point and circle**: The point is \( (2, 3) \) and we rewrite the circle as \( (x + 3)^2 + (y + 4)^2 = 121 \).
2. **Identify coefficients**: Here, \( g = 3 \), \( f = 4 \), and \( c = -96 \).
3. **Use the polar equation formula**:
\[
x \cdot 2 + y \cdot 3 + 3(x + 2) + 4(y + 3) - 96 = 0
\]
4. **Simplifying**:
\[
2x + 3y + 3x + 6 + 4y + 12 - 96 = 0
\]
Combine like terms:
\[
5x + 7y - 78 = 0
\]
### Final Equation:
\[
5x + 7y - 78 = 0
\]
---
### (iii) For the point (4, 3) with respect to the circle \(x^2 + y^2 - 8x - 6y - 9 = 0\):
1. **Identify the point and circle**: The point is \( (4, 3) \) and we rewrite the circle as \( (x - 4)^2 + (y - 3)^2 = 34 \).
2. **Identify coefficients**: Here, \( g = -4 \), \( f = -3 \), and \( c = -9 \).
3. **Use the polar equation formula**:
\[
x \cdot 4 + y \cdot 3 - 4(x - 4) - 3(y - 3) - 9 = 0
\]
4. **Simplifying**:
\[
4x + 3y - 4x + 16 - 3y + 9 - 9 = 0
\]
This simplifies to:
\[
16 = 0
\]
### Final Equation:
\[
16 = 0 \quad \text{(This indicates a special case)}
\]
---
### (iv) For the point (1, 2) with respect to the circle \(x^2 + y^2 = 7\):
1. **Identify the point and circle**: The point is \( (1, 2) \) and the circle's equation is already in standard form.
2. **Use the polar equation formula**:
\[
x \cdot 1 + y \cdot 2 = 7
\]
3. **Simplifying**:
\[
x + 2y = 7
\]
### Final Equation:
\[
x + 2y - 7 = 0
\]
---
### (v) For the point (1, -2) with respect to the circle \(x^2 + y^2 - 10x - 10y + 25 = 0\):
1. **Identify the point and circle**: The point is \( (1, -2) \) and we rewrite the circle as \( (x - 5)^2 + (y - 5)^2 = 25 \).
2. **Identify coefficients**: Here, \( g = -5 \), \( f = -5 \), and \( c = 25 \).
3. **Use the polar equation formula**:
\[
x \cdot 1 + y \cdot (-2) - 5(x - 1) - 5(y + 2) + 25 = 0
\]
4. **Simplifying**:
\[
x - 2y - 5x + 5 - 5y - 10 + 25 = 0
\]
Combine like terms:
\[
-4x - 7y + 20 = 0
\]
### Final Equation:
\[
4x + 7y - 20 = 0
\]
---
