Find the power of the point P with respect
to the circle `S = 0` when
`P =(2,3) and S=x^2+y^2-2x+8y-23=0`

Answer

To find the power of the point \( P(2, 3) \) with respect to the circle defined by the equation \( S = x^2 + y^2 - 2x + 8y - 23 = 0 \), we will follow these steps:

### Step 1: Identify the point and the circle equation
The point \( P \) is given as \( (2, 3) \) and the equation of the circle is \( S = x^2 + y^2 - 2x + 8y - 23 = 0 \).

### Step 2: Substitute the coordinates of the point into the circle equation
We will substitute \( x = 2 \) and \( y = 3 \) into the equation \( S \):
\[ S(2, 3) = (2)^2 + (3)^2 - 2(2) + 8(3) - 23 \]

### Step 3: Calculate each term
Calculating each term:
- \( (2)^2 = 4 \)
- \( (3)^2 = 9 \)
- \( -2(2) = -4 \)
- \( 8(3) = 24 \)

Now substituting these values back into the equation:
\[ S(2, 3) = 4 + 9 - 4 + 24 - 23 \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ S(2, 3) = 4 + 9 = 13 \]
\[ 13 - 4 = 9 \]
\[ 9 + 24 = 33 \]
\[ 33 - 23 = 10 \]

### Step 5: Calculate the power of the point
The power of the point \( P \) with respect to the circle is given by:
\[ \text{Power} = S(2, 3) = 10 \]

### Step 6: Final result
Thus, the power of the point \( P(2, 3) \) with respect to the circle is:
\[ \text{Power} = 10 \]