The centre of similutude of the circles `x^(2)+y^(2)-2x-6y+6=0, x^(2)+y^(2)=1` is

Answer

To find the center of similitude of the circles given by the equations \(x^2 + y^2 - 2x - 6y + 6 = 0\) and \(x^2 + y^2 = 1\), we will follow these steps:

### Step 1: Identify the first circle
The first circle is given by the equation:
\[ x^2 + y^2 - 2x - 6y + 6 = 0 \]
We can rewrite this in the standard form of a circle \(x^2 + y^2 + 2gx + 2fy + c = 0\).

### Step 2: Find the center and radius of the first circle
From the equation, we can identify:
- \(g = -1\)
- \(f = -3\)
- \(c = 6\)

The center \((h_1, k_1)\) of the first circle is given by:
\[ (h_1, k_1) = (-g, -f) = (1, 3) \]

To find the radius \(r_1\):
\[ r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + (-3)^2 - 6} = \sqrt{1 + 9 - 6} = \sqrt{4} = 2 \]

### Step 3: Identify the second circle
The second circle is given by:
\[ x^2 + y^2 = 1 \]
This can be rewritten as:
\[ x^2 + y^2 + 0x + 0y - 1 = 0 \]
Here, we have:
- \(g = 0\)
- \(f = 0\)
- \(c = -1\)

The center \((h_2, k_2)\) of the second circle is:
\[ (h_2, k_2) = (-g, -f) = (0, 0) \]

The radius \(r_2\) is:
\[ r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{0^2 + 0^2 - (-1)} = \sqrt{1} = 1 \]

### Step 4: Use the formula for the center of similitude
The center of similitude \((x, y)\) of two circles can be calculated using the formula:
\[ x = \frac{r_1 h_2 + r_2 h_1}{r_1 + r_2}, \quad y = \frac{r_1 k_2 + r_2 k_1}{r_1 + r_2} \]

Substituting the values we found:
- \(r_1 = 2\), \(r_2 = 1\)
- \(h_1 = 1\), \(k_1 = 3\)
- \(h_2 = 0\), \(k_2 = 0\)

Calculating \(x\):
\[ x = \frac{2 \cdot 0 + 1 \cdot 1}{2 + 1} = \frac{0 + 1}{3} = \frac{1}{3} \]

Calculating \(y\):
\[ y = \frac{2 \cdot 0 + 1 \cdot 3}{2 + 1} = \frac{0 + 3}{3} = 1 \]

### Final Answer
The center of similitude of the two circles is:
\[ \left(\frac{1}{3}, 1\right) \]