The angle of intersection of the two circles `x^2 + y^2 - 2x - 2y = 0 and x^2 + y^2 = 4` , is

Options

A. `30^@`
B. `60^@`
C. `90^@`
D. `45^@`

Answer

To find the angle of intersection of the two circles given by the equations \( x^2 + y^2 - 2x - 2y = 0 \) and \( x^2 + y^2 = 4 \), we can follow these steps:

### Step 1: Rewrite the equations of the circles in standard form

1. **First Circle**:
\[ x^2 + y^2 - 2x - 2y = 0 \]
Rearranging gives:
\[ (x^2 - 2x) + (y^2 - 2y) = 0 \]
Completing the square:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 0 \implies (x - 1)^2 + (y - 1)^2 = 2 \]
This represents a circle centered at \( (1, 1) \) with radius \( \sqrt{2} \).

2. **Second Circle**:
\[ x^2 + y^2 = 4 \]
This is already in standard form, representing a circle centered at \( (0, 0) \) with radius \( 2 \).

### Step 2: Identify the parameters for the angle of intersection formula

For two circles given by the general equation \( x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 \) and \( x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0 \), we have:

- For the first circle:
- \( g_1 = -1 \)
- \( f_1 = -1 \)
- \( c_1 = -2 \)

- For the second circle:
- \( g_2 = 0 \)
- \( f_2 = 0 \)
- \( c_2 = -4 \)

### Step 3: Use the formula for the angle of intersection

The angle \( \theta \) between the two circles can be calculated using the formula:
\[ \cos \theta = \frac{2(g_1g_2 + f_1f_2) - (c_1 + c_2)}{2\sqrt{(g_1^2 + f_1^2 - c_1)(g_2^2 + f_2^2 - c_2)}} \]

Substituting the values:
\[ \cos \theta = \frac{2((-1)(0) + (-1)(0)) - (-2 - 4)}{2\sqrt{((-1)^2 + (-1)^2 - (-2))(0^2 + 0^2 - (-4))}} \]
This simplifies to:
\[ \cos \theta = \frac{0 + 6}{2\sqrt{(1 + 1 + 2)(0 + 0 + 4)}} \]
\[ = \frac{6}{2\sqrt{4 \cdot 4}} = \frac{6}{2 \cdot 4} = \frac{6}{8} = \frac{3}{4} \]

### Step 4: Find the angle \( \theta \)

Now, we need to find \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{3}{4}\right) \]

Calculating \( \theta \) gives us the angle of intersection between the two circles.

### Final Result

The angle of intersection of the two circles is:
\[ \theta \approx 41.41^\circ \]