Find the equation to the circle which
touches each axis and is of radius a,

Answer

To find the equation of a circle that touches each axis and has a radius \( a \), we can follow these steps:

### Step 1: Understand the Circle's Properties
A circle that touches both the x-axis and y-axis must have its center at a distance equal to its radius from both axes. Since the radius is \( a \), the center of the circle will be at coordinates \( (h, k) \) where both \( h \) and \( k \) are equal to \( a \) or \( -a \).

### Step 2: Determine Possible Centers
The center of the circle can be in any of the four quadrants:
1. First Quadrant: \( (a, a) \)
2. Second Quadrant: \( (-a, a) \)
3. Third Quadrant: \( (-a, -a) \)
4. Fourth Quadrant: \( (a, -a) \)

### Step 3: Write the General Equation of the Circle
The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
In our case, \( r = a \).

### Step 4: Write Equations for Each Quadrant
Now we can write the equations for the circle in each quadrant based on the centers we identified:

1. **First Quadrant**: Center \( (a, a) \)
\[ (x - a)^2 + (y - a)^2 = a^2 \]

2. **Second Quadrant**: Center \( (-a, a) \)
\[ (x + a)^2 + (y - a)^2 = a^2 \]

3. **Third Quadrant**: Center \( (-a, -a) \)
\[ (x + a)^2 + (y + a)^2 = a^2 \]

4. **Fourth Quadrant**: Center \( (a, -a) \)
\[ (x - a)^2 + (y + a)^2 = a^2 \]

### Final Result
Thus, the four equations of the circles that touch each axis and have a radius \( a \) are:
1. \( (x - a)^2 + (y - a)^2 = a^2 \)
2. \( (x + a)^2 + (y - a)^2 = a^2 \)
3. \( (x + a)^2 + (y + a)^2 = a^2 \)
4. \( (x - a)^2 + (y + a)^2 = a^2 \)