Find the power of the point P w.r.t the circle S=0 when
(i) P(1,2) and `S=x^(2)+y^(2)+6x+8y-96`
(ii) P(5,-6) and `S=x^(2)+y^(2)+8x+12y+15`
(iii) P(2,4) and `S=x^(2)+y^(2)-4x-6y-12`

Answer

To find the power of a point \( P \) with respect to a circle defined by the equation \( S = 0 \), we can use the formula:

\[ \text{Power of point } P = S_1 = x^2 + y^2 + 2gx + 2fy + c \]

where \( g \), \( f \), and \( c \) are the coefficients from the general equation of the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \).

Let's solve each part step by step.

### Part (i): \( P(1, 2) \) and \( S = x^2 + y^2 + 6x + 8y - 96 \)

1. **Identify coefficients**: From the equation \( S = x^2 + y^2 + 6x + 8y - 96 \), we have:
- \( g = 3 \) (since \( 2g = 6 \))
- \( f = 4 \) (since \( 2f = 8 \))
- \( c = -96 \)

2. **Substitute point \( P(1, 2) \)**:
\[ S_1 = 1^2 + 2^2 + 6 \cdot 1 + 8 \cdot 2 - 96 \]
\[ = 1 + 4 + 6 + 16 - 96 \]
\[ = 27 - 96 = -69 \]

3. **Calculate power**:
\[ \text{Power of } P = \sqrt{S_1} = \sqrt{-69} = i\sqrt{69} \]

### Part (ii): \( P(5, -6) \) and \( S = x^2 + y^2 + 8x + 12y + 15 \)

1. **Identify coefficients**: From the equation \( S = x^2 + y^2 + 8x + 12y + 15 \), we have:
- \( g = 4 \) (since \( 2g = 8 \))
- \( f = 6 \) (since \( 2f = 12 \))
- \( c = 15 \)

2. **Substitute point \( P(5, -6) \)**:
\[ S_1 = 5^2 + (-6)^2 + 8 \cdot 5 + 12 \cdot (-6) + 15 \]
\[ = 25 + 36 + 40 - 72 + 15 \]
\[ = 101 - 72 = 29 \]

3. **Calculate power**:
\[ \text{Power of } P = \sqrt{S_1} = \sqrt{29} \]

### Part (iii): \( P(2, 4) \) and \( S = x^2 + y^2 - 4x - 6y - 12 \)

1. **Identify coefficients**: From the equation \( S = x^2 + y^2 - 4x - 6y - 12 \), we have:
- \( g = -2 \) (since \( 2g = -4 \))
- \( f = -3 \) (since \( 2f = -6 \))
- \( c = -12 \)

2. **Substitute point \( P(2, 4) \)**:
\[ S_1 = 2^2 + 4^2 - 4 \cdot 2 - 6 \cdot 4 - 12 \]
\[ = 4 + 16 - 8 - 24 - 12 \]
\[ = 20 - 44 = -24 \]

3. **Calculate power**:
\[ \text{Power of } P = \sqrt{S_1} = \sqrt{-24} = i\sqrt{24} = 2i\sqrt{6} \]

### Summary of Results:
1. Power of \( P(1, 2) \) is \( i\sqrt{69} \).
2. Power of \( P(5, -6) \) is \( \sqrt{29} \).
3. Power of \( P(2, 4) \) is \( 2i\sqrt{6} \).