The angle between the circles `x^2+y^2+4x+8y+18=0` and `x^2+y^2+2x+6y+8=0` is

Options

A. `pi/4`
B. `(2pi)/3`
C. `pi/6`
D. `pi/2`

Answer

To find the angle between the circles given by the equations \(x^2 + y^2 + 4x + 8y + 18 = 0\) and \(x^2 + y^2 + 2x + 6y + 8 = 0\), we will follow these steps:

### Step 1: Rewrite the equations in standard form
We start by rewriting each circle's equation in standard form \((x - h)^2 + (y - k)^2 = r^2\).

**For the first circle:**
\[ x^2 + y^2 + 4x + 8y + 18 = 0 \]
Completing the square:
- For \(x\): \(x^2 + 4x = (x + 2)^2 - 4\)
- For \(y\): \(y^2 + 8y = (y + 4)^2 - 16\)

Thus, we have:
\[ (x + 2)^2 - 4 + (y + 4)^2 - 16 + 18 = 0 \]
\[ (x + 2)^2 + (y + 4)^2 - 2 = 0 \implies (x + 2)^2 + (y + 4)^2 = 2 \]
This gives us the center \(C_1(-2, -4)\) and radius \(R_1 = \sqrt{2}\).

**For the second circle:**
\[ x^2 + y^2 + 2x + 6y + 8 = 0 \]
Completing the square:
- For \(x\): \(x^2 + 2x = (x + 1)^2 - 1\)
- For \(y\): \(y^2 + 6y = (y + 3)^2 - 9\)

Thus, we have:
\[ (x + 1)^2 - 1 + (y + 3)^2 - 9 + 8 = 0 \]
\[ (x + 1)^2 + (y + 3)^2 - 2 = 0 \implies (x + 1)^2 + (y + 3)^2 = 2 \]
This gives us the center \(C_2(-1, -3)\) and radius \(R_2 = \sqrt{2}\).

### Step 2: Calculate the distance between the centers
The distance \(C_1C_2\) between the centers \(C_1(-2, -4)\) and \(C_2(-1, -3)\) is calculated using the distance formula:
\[ C_1C_2 = \sqrt{((-1) - (-2))^2 + ((-3) - (-4))^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{2} \]

### Step 3: Use the angle formula
The angle \(\theta\) between the two circles can be found using the formula:
\[ \cos \theta = \frac{R_1^2 + R_2^2 - (C_1C_2)^2}{2R_1R_2} \]
Substituting the values:
- \(R_1 = R_2 = \sqrt{2}\)
- \(C_1C_2 = \sqrt{2}\)

Calculating:
\[ \cos \theta = \frac{(\sqrt{2})^2 + (\sqrt{2})^2 - (\sqrt{2})^2}{2 \cdot \sqrt{2} \cdot \sqrt{2}} = \frac{2 + 2 - 2}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 4: Find the angle \(\theta\)
From \(\cos \theta = \frac{1}{2}\), we find:
\[ \theta = \frac{\pi}{3} \text{ or } 60^\circ \]

### Conclusion
Thus, the angle between the circles is \(\frac{2\pi}{3}\) (since we are considering the obtuse angle).

### Final Answer
The angle between the circles is \(\frac{2\pi}{3}\).

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