To find the equation of the circle with a radius of 5 that touches the circle given by the equation \(x^2 + y^2 - 2x - 4y - 20 = 0\) externally at the point \((5, 5)\), we can follow these steps:
### Step 1: Rewrite the given circle's equation in standard form
We start with the equation of the first circle:
\[
x^2 + y^2 - 2x - 4y - 20 = 0
\]
To convert this into standard form, we complete the square for both \(x\) and \(y\).
1. For \(x\):
\[
x^2 - 2x \quad \text{can be rewritten as} \quad (x - 1)^2 - 1
\]
2. For \(y\):
\[
y^2 - 4y \quad \text{can be rewritten as} \quad (y - 2)^2 - 4
\]
Now substituting these back into the equation:
\[
(x - 1)^2 - 1 + (y - 2)^2 - 4 - 20 = 0
\]
This simplifies to:
\[
(x - 1)^2 + (y - 2)^2 - 25 = 0
\]
Thus, we have:
\[
(x - 1)^2 + (y - 2)^2 = 25
\]
This shows that the center of the first circle is \((1, 2)\) and the radius is \(5\).
### Step 2: Determine the center of the second circle
Let the center of the second circle be \((h, k)\) and its radius is given as \(5\). Since the two circles touch externally at the point \((5, 5)\), the distance between their centers must equal the sum of their radii.
The distance between the centers is given by:
\[
\sqrt{(h - 1)^2 + (k - 2)^2}
\]
Since both circles have a radius of \(5\), the distance must equal:
\[
5 + 5 = 10
\]
Thus, we have:
\[
\sqrt{(h - 1)^2 + (k - 2)^2} = 10
\]
Squaring both sides gives:
\[
(h - 1)^2 + (k - 2)^2 = 100
\]
### Step 3: Use the point of tangency to find \(h\) and \(k\)
Since the circles touch at \((5, 5)\), we can use the midpoint formula. The point \((5, 5)\) is the midpoint between the centers \((1, 2)\) and \((h, k)\):
\[
\left(\frac{1 + h}{2}, \frac{2 + k}{2}\right) = (5, 5)
\]
From this, we can set up two equations:
1. \(\frac{1 + h}{2} = 5\)
2. \(\frac{2 + k}{2} = 5\)
Solving these equations:
1. \(1 + h = 10 \Rightarrow h = 9\)
2. \(2 + k = 10 \Rightarrow k = 8\)
Thus, the center of the second circle is \((9, 8)\).
### Step 4: Write the equation of the second circle
Now that we have the center \((9, 8)\) and the radius \(5\), we can write the equation of the second circle in standard form:
\[
(x - 9)^2 + (y - 8)^2 = 5^2
\]
This simplifies to:
\[
(x - 9)^2 + (y - 8)^2 = 25
\]
### Final Answer
The equation of the circle is:
\[
(x - 9)^2 + (y - 8)^2 = 25
\]
