Circles `x^(2) + y^(2) - 2x = 0 and x^(2) + y^(2) + 6x - 6y + 2 = 0` touch each other extermally. Then point of contact is

Options

A. `((3)/(5),(5)/(3))`
B. `((3)/(5),(1)/(5))`
C. `((1)/(5),(1)/(5))`
D. `((1)/(5),(3)/(5))`

Answer

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

### Step 1: Rewrite the equations of the circles
The first circle can be rewritten as:
\[ x^2 + y^2 - 2x = 0 \implies (x - 1)^2 + y^2 = 1^2 \]
This indicates that the center \( C_1 \) is at \( (1, 0) \) and the radius \( r_1 = 1 \).

The second circle can be rewritten as:
\[ x^2 + y^2 + 6x - 6y + 2 = 0 \implies (x + 3)^2 + (y - 3)^2 = 4^2 \]
This indicates that the center \( C_2 \) is at \( (-3, 3) \) and the radius \( r_2 = 4 \).

### Step 2: Calculate the distance between the centers
To find the distance \( d \) between the centers \( C_1 \) and \( C_2 \):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{((-3) - 1)^2 + (3 - 0)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Verify the circles touch externally
For two circles to touch externally, the distance \( d \) between their centers must equal the sum of their radii:
\[ d = r_1 + r_2 = 1 + 4 = 5 \]
Since \( d = 5 \), the circles touch each other externally.

### Step 4: Find the point of contact
Let the point of contact be \( P \). The point \( P \) divides the line segment \( C_1C_2 \) in the ratio of the radii \( r_1 : r_2 = 1 : 4 \).

Using the section formula, the coordinates of point \( P \) can be calculated as:
\[ P = \left( \frac{4 \cdot x_1 + 1 \cdot x_2}{r_1 + r_2}, \frac{4 \cdot y_1 + 1 \cdot y_2}{r_1 + r_2} \right) \]
Substituting the coordinates of \( C_1(1, 0) \) and \( C_2(-3, 3) \):
\[ P = \left( \frac{4 \cdot 1 + 1 \cdot (-3)}{1 + 4}, \frac{4 \cdot 0 + 1 \cdot 3}{1 + 4} \right) = \left( \frac{4 - 3}{5}, \frac{0 + 3}{5} \right) = \left( \frac{1}{5}, \frac{3}{5} \right) \]

### Final Answer
The point of contact is:
\[ P = \left( \frac{1}{5}, \frac{3}{5} \right) \]