If one of the diameters of the curve `""^(2) + y ^(2) - 4x - 6y +9=0` is a chord of a circle with centre `(1,1)` then the radius of this circel is

Options

A. `sqrt(3)`
B. `sqrt(2)`
C. 3
D. 2

Answer

To solve the problem step by step, we will first analyze the given equation of the circle and then find the required radius of the second circle.

### Step 1: Rewrite the equation of the circle
The given equation is:
\[ x^2 + y^2 - 4x - 6y + 9 = 0 \]
We will rearrange this equation to find the center and radius of the circle.

### Step 2: Complete the square
To find the center and radius, we complete the square for both \(x\) and \(y\).

1. For \(x\):
\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \]

2. For \(y\):
\[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \]

Now substituting back into the equation:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 + 9 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 3)^2 - 4 = 0 \]
So:
\[ (x - 2)^2 + (y - 3)^2 = 4 \]

### Step 3: Identify the center and radius
From the equation \((x - 2)^2 + (y - 3)^2 = 4\), we can identify:
- Center: \(C(2, 3)\)
- Radius: \(r = \sqrt{4} = 2\)

### Step 4: Find the diameter of the circle
The diameter \(d\) of the circle is:
\[ d = 2 \times r = 2 \times 2 = 4 \]

### Step 5: Determine the length of the chord
Since one of the diameters of the circle is a chord of the second circle with center at \(O(1, 1)\), we can denote the endpoints of the diameter as \(A\) and \(B\). The length of the chord (which is the diameter) is \(4\).

### Step 6: Use the right triangle relationship
The line from the center \(O(1, 1)\) to the midpoint \(M\) of the chord \(AB\) will be perpendicular to the chord. The distance \(OM\) can be calculated using the distance formula.

### Step 7: Calculate the distance \(CO\)
The distance \(CO\) from center \(C(2, 3)\) to center \(O(1, 1)\) is:
\[ CO = \sqrt{(1 - 2)^2 + (1 - 3)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 8: Apply the Pythagorean theorem
In triangle \(AOC\) (where \(A\) and \(B\) are the endpoints of the diameter):
\[ AC^2 = AO^2 + CO^2 \]
Where \(AO\) is half the diameter (which is \(2\)):
\[ AC^2 = 2^2 + (\sqrt{5})^2 \]
\[ AC^2 = 4 + 5 = 9 \]
So:
\[ AC = \sqrt{9} = 3 \]

### Conclusion: Radius of the second circle
Thus, the radius \(R\) of the second circle is:
\[ R = AC = 3 \]

### Final Answer
The radius of the circle with center \((1, 1)\) is \(3\).

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