Find the angle between the circles `S:x^(2)+y^(2)-4x+6y+11=0andS':x^(2)+y^(2)-2x+8y+13=0`

Answer

To find the angle between the circles given by the equations \( S: x^2 + y^2 - 4x + 6y + 11 = 0 \) and \( S': x^2 + y^2 - 2x + 8y + 13 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The general form of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the first circle \( S \):
- Coefficients: \( 2g_1 = -4 \), \( 2f_1 = 6 \), \( c_1 = 11 \)
- Thus, \( g_1 = -2 \), \( f_1 = 3 \)

From the second circle \( S' \):
- Coefficients: \( 2g_2 = -2 \), \( 2f_2 = 8 \), \( c_2 = 13 \)
- Thus, \( g_2 = -1 \), \( f_2 = 4 \)

### Step 2: Find the centers and radii of the circles
The center of circle \( S \) is given by \( C_1 = (-g_1, -f_1) = (2, -3) \).

The radius \( r_1 \) is calculated as:
\[ r_1 = \sqrt{g_1^2 + f_1^2 - c_1} = \sqrt{(-2)^2 + 3^2 - 11} = \sqrt{4 + 9 - 11} = \sqrt{2} \]

The center of circle \( S' \) is given by \( C_2 = (-g_2, -f_2) = (1, -4) \).

The radius \( r_2 \) is calculated as:
\[ r_2 = \sqrt{g_2^2 + f_2^2 - c_2} = \sqrt{(-1)^2 + 4^2 - 13} = \sqrt{1 + 16 - 13} = \sqrt{4} = 2 \]

### Step 3: Find the distance between the centers
The distance \( d \) between the centers \( C_1 \) and \( C_2 \) is calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(1 - 2)^2 + (-4 + 3)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 4: Use the formula for the angle between two circles
The angle \( \theta \) between two circles can be found using the cosine formula:
\[ \cos \theta = \frac{r_1^2 + r_2^2 - d^2}{2r_1r_2} \]
Substituting the values:
- \( r_1^2 = (\sqrt{2})^2 = 2 \)
- \( r_2^2 = 2^2 = 4 \)
- \( d^2 = (\sqrt{2})^2 = 2 \)

Now substituting these into the formula:
\[ \cos \theta = \frac{2 + 4 - 2}{2 \cdot \sqrt{2} \cdot 2} = \frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 5: Find the angle \( \theta \)
Since \( \cos \theta = \frac{1}{\sqrt{2}} \), we find:
\[ \theta = 45^\circ \]

Thus, the angle between the circles is \( 45^\circ \).