The circles `x^(2)+ y^(2) -6x-2y +9 = 0 ` and `x^(2) + y^(2) =18` are such that they `:`

Options

A. touch each other
B. intersect
C. one lies inside the other
D. None of these

Answer

To determine the relationship between the two given circles, we need to analyze their equations and find their centers and radii.

### Step 1: Rewrite the first circle's equation
The first circle is given by the equation:
\[ x^2 + y^2 - 6x - 2y + 9 = 0 \]

We can rearrange this into standard form by completing the square.

1. Group the \(x\) and \(y\) terms:
\[ (x^2 - 6x) + (y^2 - 2y) + 9 = 0 \]

2. Complete the square for \(x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]

3. Complete the square for \(y\):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

4. Substitute back into the equation:
\[ (x - 3)^2 - 9 + (y - 1)^2 - 1 + 9 = 0 \]
\[ (x - 3)^2 + (y - 1)^2 - 1 = 0 \]
\[ (x - 3)^2 + (y - 1)^2 = 1 \]

### Step 2: Identify the center and radius of the first circle
From the equation \((x - 3)^2 + (y - 1)^2 = 1\):
- Center: \(C_1(3, 1)\)
- Radius: \(r_1 = \sqrt{1} = 1\)

### Step 3: Rewrite the second circle's equation
The second circle is given by:
\[ x^2 + y^2 = 18 \]

This can be rewritten in standard form as:
\[ (x - 0)^2 + (y - 0)^2 = 18 \]

### Step 4: Identify the center and radius of the second circle
From the equation \((x - 0)^2 + (y - 0)^2 = 18\):
- Center: \(C_2(0, 0)\)
- Radius: \(r_2 = \sqrt{18} = 3\sqrt{2}\)

### Step 5: Determine the distance between the centers
To find the relationship between the two circles, we need to calculate the distance \(d\) between their centers \(C_1(3, 1)\) and \(C_2(0, 0)\):
\[ d = \sqrt{(3 - 0)^2 + (1 - 0)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]

### Step 6: Compare the distance with the sum and difference of the radii
- Sum of the radii: \(r_1 + r_2 = 1 + 3\sqrt{2}\)
- Difference of the radii: \(r_2 - r_1 = 3\sqrt{2} - 1\)

### Step 7: Analyze the relationship
1. If \(d > r_1 + r_2\), the circles are separate.
2. If \(d = r_1 + r_2\), the circles touch externally.
3. If \(d < r_1 + r_2\) and \(d > |r_2 - r_1|\), the circles intersect.
4. If \(d = |r_2 - r_1|\), the circles touch internally.
5. If \(d < |r_2 - r_1|\), one circle lies inside the other.

### Conclusion
Since \(\sqrt{10} < 1 + 3\sqrt{2}\) and \(\sqrt{10} < 3\sqrt{2} - 1\), we conclude that one circle lies inside the other.

### Final Answer
The circles are such that one lies inside the other.

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