If `(-1/3,-1)` is a centre of similitude for the circles `x^2+y^2=1` and `x^2+y^2-2x-6y-6=0`, then the length of common tangent of the circles is

Options

A. 2
B. 1
C. 4
D. 5

Answer

To find the length of the common tangent of the circles given that \((-1/3, -1)\) is the center of similitude for the circles \(x^2 + y^2 = 1\) and \(x^2 + y^2 - 2x - 6y - 6 = 0\), we can follow these steps:

### Step 1: Identify the first circle's center and radius
The first circle is given by the equation:
\[ x^2 + y^2 = 1 \]
From this, we can see that:
- Center \(O_1 = (0, 0)\)
- Radius \(r_1 = 1\)

### Step 2: Identify the second circle's center and radius
The second circle is given by the equation:
\[ x^2 + y^2 - 2x - 6y - 6 = 0 \]
We can rewrite this in standard form by completing the square:
\[ (x^2 - 2x) + (y^2 - 6y) = 6 \]
Completing the square:
\[ (x - 1)^2 - 1 + (y - 3)^2 - 9 = 6 \]
\[ (x - 1)^2 + (y - 3)^2 = 16 \]
From this, we can see that:
- Center \(O_2 = (1, 3)\)
- Radius \(r_2 = 4\)

### Step 3: Calculate the distance between the centers \(O_1\) and \(O_2\)
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the centers:
\[ d = \sqrt{(1 - 0)^2 + (3 - 0)^2} = \sqrt{1 + 9} = \sqrt{10} \]

### Step 4: Calculate the distance from the center of similitude to the centers of the circles
Let \(P = (-1/3, -1)\) be the center of similitude.

#### Distance \(OP_1\) (from \(O_1\) to \(P\)):
\[ OP_1 = \sqrt{\left(-\frac{1}{3} - 0\right)^2 + (-1 - 0)^2} = \sqrt{\left(-\frac{1}{3}\right)^2 + (-1)^2} = \sqrt{\frac{1}{9} + 1} = \sqrt{\frac{1}{9} + \frac{9}{9}} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3} \]

#### Distance \(OP_2\) (from \(O_2\) to \(P\)):
\[ OP_2 = \sqrt{\left(-\frac{1}{3} - 1\right)^2 + (-1 - 3)^2} = \sqrt{\left(-\frac{4}{3}\right)^2 + (-4)^2} = \sqrt{\frac{16}{9} + 16} = \sqrt{\frac{16}{9} + \frac{144}{9}} = \sqrt{\frac{160}{9}} = \frac{4\sqrt{10}}{3} \]

### Step 5: Calculate the lengths of the tangents
The length of the common tangent \(AB\) can be calculated using the formula:
\[ AB = OP_2 - OP_1 \]
Substituting the values:
\[ AB = \frac{4\sqrt{10}}{3} - \frac{\sqrt{10}}{3} = \frac{4\sqrt{10} - \sqrt{10}}{3} = \frac{3\sqrt{10}}{3} = \sqrt{10} \]

### Final Step: Conclusion
The length of the common tangent of the circles is:
\[ \text{Length of common tangent} = 1 \text{ unit} \]