`S:x ^(2) + y ^(2) -8x + 10y =0 and L : x -y -9=0` are the equations of a circle and a line.

Options

A. L is a normal to the circle S.
B. S is the only circle having radius `sqrt41` and a diameter along L.
C. L is a tangent to the circle S.
D. L does not intersect the circle S.

Answer

To solve the problem, we need to analyze the given equations of the circle and the line step by step.

### Step 1: Identify the Circle's Equation
The equation of the circle is given as:
\[ S: x^2 + y^2 - 8x + 10y = 0 \]

### Step 2: Rewrite the Circle's Equation
We can rewrite the equation of the circle in standard form by completing the square for both \(x\) and \(y\).

1. Rearranging the equation:
\[ x^2 - 8x + y^2 + 10y = 0 \]

2. Completing the square for \(x\):
\[ x^2 - 8x = (x - 4)^2 - 16 \]

3. Completing the square for \(y\):
\[ y^2 + 10y = (y + 5)^2 - 25 \]

4. Substituting back into the equation:
\[ (x - 4)^2 - 16 + (y + 5)^2 - 25 = 0 \]
\[ (x - 4)^2 + (y + 5)^2 = 41 \]

### Step 3: Identify the Center and Radius of the Circle
From the standard form of the circle's equation \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center \((h, k) = (4, -5)\)
- Radius \(r = \sqrt{41}\)

### Step 4: Identify the Line's Equation
The equation of the line is given as:
\[ L: x - y - 9 = 0 \]
This can be rewritten as:
\[ y = x - 9 \]

### Step 5: Check if the Center of the Circle Lies on the Line
To check if the center of the circle lies on the line, substitute the center \((4, -5)\) into the line's equation:
\[ 4 - (-5) - 9 = 0 \]
\[ 4 + 5 - 9 = 0 \]
\[ 0 = 0 \]
This means the center \((4, -5)\) lies on the line \(L\).

### Step 6: Determine the Relationship Between the Circle and the Line
Since the center of the circle lies on the line, the line is not a tangent but rather a normal to the circle. This means the line passes through the center of the circle.

### Step 7: Conclusion
Based on the analysis:
- Option 1 is correct: The line \(L\) is a normal to the circle \(S\).
- Option 2 is incorrect: There is not only one circle with radius \(\sqrt{41}\) and diameter along line \(L\).
- Option 3 is incorrect: The line is not a tangent.
- Option 4 is incorrect: The line does intersect the circle.