Find the equation of the radical axis of the following circles.
`x^2+y^2+2x+4y+1=0`
`x^2+y^2+4x+y=0`

Answer

To find the equation of the radical axis of the given circles, we will follow these steps:

1. **Write down the equations of the circles**:
The equations of the circles are:
- Circle 1: \( S_1: x^2 + y^2 + 2x + 4y + 1 = 0 \)
- Circle 2: \( S_2: x^2 + y^2 + 4x + y = 0 \)

2. **Set up the equation for the radical axis**:
The equation of the radical axis is given by setting \( S_1 = S_2 \):
\[ x^2 + y^2 + 2x + 4y + 1 = x^2 + y^2 + 4x + y \]

3. **Cancel out common terms**:
Subtract \( x^2 + y^2 \) from both sides:
\[ 2x + 4y + 1 = 4x + y \]

4. **Rearrange the equation**:
Move all terms to one side:
\[ 2x + 4y + 1 - 4x - y = 0 \]
Simplifying this gives:
\[ -2x + 3y + 1 = 0 \]

5. **Rearrange to standard form**:
Rearranging gives:
\[ 2x - 3y = -1 \]

6. **Final equation of the radical axis**:
Thus, the equation of the radical axis is:
\[ 2x - 3y = 1 \]