The angle between the circles `x^2+y^2-2x-4y+3=0` and `x^2+y^2-4x-6y+11=0` is

Options

A. `pi/3`
B. `pi/6`
C. `pi/2`
D. `(5pi)/3`

Answer

To find the angle between the two circles given by the equations \( x^2 + y^2 - 2x - 4y + 3 = 0 \) and \( x^2 + y^2 - 4x - 6y + 11 = 0 \), we will follow these steps:

### Step 1: Rewrite the equations in standard form
The general form of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
We will rewrite both equations to identify the center and radius.

**For the first circle:**
\[ x^2 + y^2 - 2x - 4y + 3 = 0 \]
Comparing with the general form, we have:
- \( 2g = -2 \) → \( g = -1 \)
- \( 2f = -4 \) → \( f = -2 \)
- \( c = 3 \)

The center \( C_1 \) is given by \( (-g, -f) = (1, 2) \) and the radius \( r_1 \) is calculated as:
\[ r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + (-2)^2 - 3} = \sqrt{1 + 4 - 3} = \sqrt{2} \]

**For the second circle:**
\[ x^2 + y^2 - 4x - 6y + 11 = 0 \]
Comparing with the general form, we have:
- \( 2g = -4 \) → \( g = -2 \)
- \( 2f = -6 \) → \( f = -3 \)
- \( c = 11 \)

The center \( C_2 \) is given by \( (-g, -f) = (2, 3) \) and the radius \( r_2 \) is calculated as:
\[ r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-3)^2 - 11} = \sqrt{4 + 9 - 11} = \sqrt{2} \]

### Step 2: Calculate the distance between the centers
The distance \( d \) between the centers \( C_1(1, 2) \) and \( C_2(2, 3) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 1)^2 + (3 - 2)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 3: Use the formula for the angle between two circles
The angle \( \theta \) between two circles can be found using the formula:
\[ \cos \theta = \frac{d^2 - r_1^2 - r_2^2}{2 r_1 r_2} \]
Substituting the values we have:
- \( d^2 = (\sqrt{2})^2 = 2 \)
- \( r_1^2 = (\sqrt{2})^2 = 2 \)
- \( r_2^2 = (\sqrt{2})^2 = 2 \)

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

### Step 4: Find the angle \( \theta \)
Since \( \cos \theta = -\frac{1}{2} \), we find:
\[ \theta = \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \text{ or } \frac{4\pi}{3} \]
However, the angle between circles is typically taken as the acute angle, which is:
\[ \theta = \frac{\pi}{3} \]

### Final Answer
The angle between the circles is \( \frac{\pi}{3} \).

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