Consider the circle `x^2+y^2 -8x-18y +93=0` with the center C and a point `P(2,5)` out side it. From P a pair of tangents PQ and PR are drawn to the circle with S as mid point of QR. The line joining P to C intersects the given circle at A and B. Which of the following hold (s)

Options

A. CP is the arithmetic mean of AP and BP
B. PR is the geometric mean of PS and PC
C. PS is the harmonic mean of PA and PB
D. The angle between the two tangents from P is `tan^(-1)((4)/(3))`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 - 8x - 18y + 93 = 0 \]

We can rewrite this in standard form by completing the square.

### Step 2: Completing the Square
1. For \(x\):
\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \]
2. For \(y\):
\[ y^2 - 18y \rightarrow (y - 9)^2 - 81 \]

Now substituting back into the equation:
\[ (x - 4)^2 - 16 + (y - 9)^2 - 81 + 93 = 0 \]
This simplifies to:
\[ (x - 4)^2 + (y - 9)^2 - 4 = 0 \]
Thus, we have:
\[ (x - 4)^2 + (y - 9)^2 = 2^2 \]

### Step 3: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\):
- Center \(C\) is \((4, 9)\)
- Radius \(r = 2\)

### Step 4: Calculate Distance \(CP\)
The point \(P\) is given as \((2, 5)\). We will use the distance formula to find \(CP\):
\[ CP = \sqrt{(4 - 2)^2 + (9 - 5)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \]

### Step 5: Calculate Distances \(AP\) and \(BP\)
Using the property of tangents from an external point:
- The distance from point \(P\) to the points of tangency \(Q\) and \(R\) can be found as:
\[ AP = CP - r = \sqrt{20} - 2 \]
\[ BP = CP + r = \sqrt{20} + 2 \]

### Step 6: Verify the Arithmetic Mean
We need to check if:
\[ CP = \frac{AP + BP}{2} \]
Substituting the values:
\[ CP = \sqrt{20} \]
\[ AP = \sqrt{20} - 2 \]
\[ BP = \sqrt{20} + 2 \]
Calculating the average:
\[ \frac{AP + BP}{2} = \frac{(\sqrt{20} - 2) + (\sqrt{20} + 2)}{2} = \frac{2\sqrt{20}}{2} = \sqrt{20} \]

### Conclusion
Since \(CP = \frac{AP + BP}{2}\), it holds true that \(CP\) is the arithmetic mean of \(AP\) and \(BP\).