Let `S_(1):x^(2)+y^(2)-2x=0andS_(2):x^(2)+y^(2)+6x-6y+2=0` Do these circles

Options

A. touch intermally
B. `S_(1)` lies completely inside the other circle `S_(2)`
C. touch externally
D. None of these

Answer

To determine the relationship between the two circles given by the equations \( S_1: x^2 + y^2 - 2x = 0 \) and \( S_2: x^2 + y^2 + 6x - 6y + 2 = 0 \), we will follow these steps:

### Step 1: Rewrite the equations in standard form
1. **For \( S_1 \)**:
\[ x^2 + y^2 - 2x = 0 \implies (x^2 - 2x + 1) + y^2 = 1 \implies (x - 1)^2 + y^2 = 1 \]
This represents a circle with center \( C_1(1, 0) \) and radius \( r_1 = 1 \).

2. **For \( S_2 \)**:
\[ x^2 + y^2 + 6x - 6y + 2 = 0 \implies (x^2 + 6x + 9) + (y^2 - 6y + 9) = 16 \implies (x + 3)^2 + (y - 3)^2 = 16 \]
This represents a circle with center \( C_2(-3, 3) \) and radius \( r_2 = 4 \).

### Step 2: Find the distance between the centers of the circles
The distance \( d \) between the centers \( C_1(1, 0) \) and \( C_2(-3, 3) \) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{((-3) - 1)^2 + (3 - 0)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Compare the distance with the sum of the radii
Now, we calculate the sum of the radii:
\[ r_1 + r_2 = 1 + 4 = 5 \]

### Step 4: Determine the relationship between the circles
Since the distance between the centers \( d = 5 \) is equal to the sum of the radii \( r_1 + r_2 = 5 \), the circles touch each other externally.

### Conclusion
Thus, the circles \( S_1 \) and \( S_2 \) touch externally.