Let `alpha` chord of a circle be that chord of the circle which subtends an angle `alpha` at the center.
The distance of `2pi //3` chord of `x^(2)+y^(2)+2x+4y+1=0` from the center is

Options

A. `(1)/(sqrt2)`
B. 1
C. `sqrt2`
D. 2

Answer

To find the distance of the chord subtending an angle of \( \frac{2\pi}{3} \) at the center of the circle given by the equation \( x^2 + y^2 + 2x + 4y + 1 = 0 \), we can follow these steps:

### Step 1: Rewrite the Circle's Equation
The equation of the circle is given as:
\[ x^2 + y^2 + 2x + 4y + 1 = 0 \]

We can rewrite this in standard form by completing the square.

### Step 2: Complete the Square
1. For \( x^2 + 2x \):
\[ x^2 + 2x = (x + 1)^2 - 1 \]

2. For \( y^2 + 4y \):
\[ y^2 + 4y = (y + 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ (x + 1)^2 - 1 + (y + 2)^2 - 4 + 1 = 0 \]
This simplifies to:
\[ (x + 1)^2 + (y + 2)^2 - 4 = 0 \]
Thus, we have:
\[ (x + 1)^2 + (y + 2)^2 = 4 \]

### Step 3: Identify the Center and Radius
From the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), we can identify:
- Center \( (h, k) = (-1, -2) \)
- Radius \( r = \sqrt{4} = 2 \)

### Step 4: Use the Distance Formula for the Chord
The distance \( d \) of a chord from the center of the circle can be calculated using the formula:
\[ d = r \cdot \cos\left(\frac{\alpha}{2}\right) \]
where \( r \) is the radius and \( \alpha \) is the angle subtended by the chord at the center.

### Step 5: Calculate \( \cos\left(\frac{\alpha}{2}\right) \)
Given \( \alpha = \frac{2\pi}{3} \):
\[ \frac{\alpha}{2} = \frac{2\pi}{6} = \frac{\pi}{3} \]
Now, we find:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 6: Substitute Values into the Distance Formula
Now substituting the values into the distance formula:
\[ d = 2 \cdot \cos\left(\frac{\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1 \]

### Final Answer
The distance of the chord subtending an angle of \( \frac{2\pi}{3} \) from the center is:
\[ \boxed{1} \]