Find the equation of the radical axis of the following circles.
`x^2+y^2-2x-4y-1=0`
`x^2+y^2-4x-6y+5=0`

Answer

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

### Step 1: Write down the equations of the circles.
The equations of the circles are:
1. \( C_1: x^2 + y^2 - 2x - 4y - 1 = 0 \)
2. \( C_2: x^2 + y^2 - 4x - 6y + 5 = 0 \)

### Step 2: Identify the general form of the circle equations.
The general form of a circle can be represented as:
\[ S: x^2 + y^2 + Dx + Ey + F = 0 \]
where \( D, E, \) and \( F \) are constants.

For \( C_1 \):
- \( D_1 = -2 \)
- \( E_1 = -4 \)
- \( F_1 = -1 \)

For \( C_2 \):
- \( D_2 = -4 \)
- \( E_2 = -6 \)
- \( F_2 = 5 \)

### Step 3: Set up the equation for the radical axis.
The equation of the radical axis is given by:
\[ S_1 - S_2 = 0 \]
This means we will equate the left-hand sides of both circle equations.

### Step 4: Write the equations of \( S_1 \) and \( S_2 \).
From the equations of the circles:
- \( S_1 = x^2 + y^2 - 2x - 4y - 1 \)
- \( S_2 = x^2 + y^2 - 4x - 6y + 5 \)

### Step 5: Set \( S_1 = S_2 \).
Now, we equate \( S_1 \) and \( S_2 \):
\[ x^2 + y^2 - 2x - 4y - 1 = x^2 + y^2 - 4x - 6y + 5 \]

### Step 6: Simplify the equation.
Cancel \( x^2 \) and \( y^2 \) from both sides:
\[ -2x - 4y - 1 = -4x - 6y + 5 \]
Rearranging gives:
\[ -2x + 4x - 4y + 6y - 1 - 5 = 0 \]
This simplifies to:
\[ 2x + 2y - 6 = 0 \]

### Step 7: Divide the equation by 2.
To simplify further, divide the entire equation by 2:
\[ x + y - 3 = 0 \]

### Final Result:
Thus, the equation of the radical axis is:
\[ \boxed{x + y = 3} \]