To find the equation of a circle that touches both the axes and has a radius of 5, we can follow these steps:
### Step 1: Understand the properties of the circle
A circle that touches both the x-axis and y-axis means that the distance from the center of the circle to both axes is equal to the radius of the circle.
### Step 2: Determine the center of the circle
Since the radius is 5, the center of the circle must be at (5, 5) in the first quadrant, where both x and y coordinates are positive.
### Step 3: Write the standard equation of the circle
The standard form of the equation of a circle with center (h, k) and radius r is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Substituting h = 5, k = 5, and r = 5 into the equation, we get:
\[
(x - 5)^2 + (y - 5)^2 = 5^2
\]
This simplifies to:
\[
(x - 5)^2 + (y - 5)^2 = 25
\]
### Step 4: Expand the equation
Now we expand the equation:
\[
(x - 5)^2 = x^2 - 10x + 25
\]
\[
(y - 5)^2 = y^2 - 10y + 25
\]
Combining these, we have:
\[
x^2 - 10x + 25 + y^2 - 10y + 25 = 25
\]
This simplifies to:
\[
x^2 + y^2 - 10x - 10y + 25 + 25 - 25 = 0
\]
Thus, we have:
\[
x^2 + y^2 - 10x - 10y + 25 = 0
\]
### Step 5: Check for other quadrants
The circle can also be in other quadrants:
1. In the second quadrant, the center would be (-5, 5).
2. In the third quadrant, the center would be (-5, -5).
3. In the fourth quadrant, the center would be (5, -5).
For each of these centers, we can derive the equations similarly:
- For center (-5, 5):
\[
(x + 5)^2 + (y - 5)^2 = 25
\]
- For center (-5, -5):
\[
(x + 5)^2 + (y + 5)^2 = 25
\]
- For center (5, -5):
\[
(x - 5)^2 + (y + 5)^2 = 25
\]
### Final Equations
1. For center (5, 5): \( x^2 + y^2 - 10x - 10y + 25 = 0 \)
2. For center (-5, 5): \( x^2 + y^2 + 10x - 10y + 25 = 0 \)
3. For center (-5, -5): \( x^2 + y^2 + 10x + 10y + 25 = 0 \)
4. For center (5, -5): \( x^2 + y^2 - 10x + 10y + 25 = 0 \)
