Find the equation of the common tangent of the following circles at their point of contact.
`x^2+y^2-8y-4=0`
`x^2+y^2-2x-4y=0`

Answer

To find the equation of the common tangent of the given circles, we will follow these steps:

### Step 1: Write the equations of the circles
The equations of the circles are given as:
1. \( S_1: x^2 + y^2 - 8y - 4 = 0 \)
2. \( S_2: x^2 + y^2 - 2x - 4y = 0 \)

### Step 2: Rearrange the equations
We can rearrange both equations to identify their centers and radii.

For \( S_1 \):
\[ x^2 + (y^2 - 8y) - 4 = 0 \implies x^2 + (y - 4)^2 - 16 - 4 = 0 \implies x^2 + (y - 4)^2 = 20 \]
This represents a circle with center \( (0, 4) \) and radius \( \sqrt{20} = 2\sqrt{5} \).

For \( S_2 \):
\[ x^2 - 2x + y^2 - 4y = 0 \implies (x^2 - 2x + 1) + (y^2 - 4y + 4) = 5 \implies (x - 1)^2 + (y - 2)^2 = 5 \]
This represents a circle with center \( (1, 2) \) and radius \( \sqrt{5} \).

### Step 3: Set up the equation for the common tangent
The equation of the common tangent to the circles can be found by equating the two circle equations:
\[ S_1 = S_2 \]
This gives us:
\[ x^2 + y^2 - 8y - 4 = x^2 + y^2 - 2x - 4y \]

### Step 4: Simplify the equation
Cancel out \( x^2 \) and \( y^2 \) from both sides:
\[ -8y - 4 = -2x - 4y \]
Rearranging gives:
\[ 2x - 8y + 4y - 4 = 0 \]
This simplifies to:
\[ 2x - 4y - 4 = 0 \]

### Step 5: Divide by 2
To simplify further, divide the entire equation by 2:
\[ x - 2y - 2 = 0 \]

### Step 6: Write the final equation
Thus, the equation of the common tangent is:
\[ x - 2y = 2 \]

### Summary of Steps:
1. Write the equations of the circles.
2. Rearrange to identify centers and radii.
3. Set the equations equal to find the common tangent.
4. Simplify the resulting equation.
5. Divide by 2 to get the final form.