To find the equation of the circle with radius \( a \) that passes through two points on the x-axis, which are at a distance \( b \) from the origin, we can follow these steps:
### Step 1: Identify the Points on the x-axis
The two points on the x-axis that are at a distance \( b \) from the origin are:
- \( (b, 0) \)
- \( (-b, 0) \)
### Step 2: Write the Standard Equation of the Circle
The standard 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, the radius \( r = a \), so the equation becomes:
\[
(x - h)^2 + (y - k)^2 = a^2
\]
### Step 3: Substitute the Points into the Circle's Equation
Since the circle passes through the points \( (b, 0) \) and \( (-b, 0) \), we can substitute these points into the circle's equation.
1. For the point \( (b, 0) \):
\[
(b - h)^2 + (0 - k)^2 = a^2
\]
This simplifies to:
\[
(b - h)^2 + k^2 = a^2 \quad \text{(Equation 1)}
\]
2. For the point \( (-b, 0) \):
\[
(-b - h)^2 + (0 - k)^2 = a^2
\]
This simplifies to:
\[
(-b - h)^2 + k^2 = a^2 \quad \text{(Equation 2)}
\]
### Step 4: Expand Both Equations
Now, we will expand both equations.
**From Equation 1**:
\[
(b - h)^2 + k^2 = a^2
\]
Expanding gives:
\[
b^2 - 2bh + h^2 + k^2 = a^2 \quad \text{(1)}
\]
**From Equation 2**:
\[
(-b - h)^2 + k^2 = a^2
\]
Expanding gives:
\[
b^2 + 2bh + h^2 + k^2 = a^2 \quad \text{(2)}
\]
### Step 5: Subtract the Two Equations
Now, we can subtract Equation 1 from Equation 2:
\[
(b^2 + 2bh + h^2 + k^2) - (b^2 - 2bh + h^2 + k^2) = 0
\]
This simplifies to:
\[
4bh = 0
\]
Since \( b \neq 0 \), we have:
\[
h = 0
\]
### Step 6: Substitute \( h \) Back to Find \( k \)
Now that we have \( h = 0 \), we can substitute this back into either Equation 1 or Equation 2 to find \( k \). Using Equation 1:
\[
(b - 0)^2 + k^2 = a^2
\]
This simplifies to:
\[
b^2 + k^2 = a^2
\]
Thus, we can solve for \( k^2 \):
\[
k^2 = a^2 - b^2
\]
### Step 7: Write the Final Equation of the Circle
Now we have \( h = 0 \) and \( k = \sqrt{a^2 - b^2} \) (considering the positive root for the y-coordinate). The equation of the circle becomes:
\[
(x - 0)^2 + \left(y - \sqrt{a^2 - b^2}\right)^2 = a^2
\]
or simply:
\[
x^2 + \left(y - \sqrt{a^2 - b^2}\right)^2 = a^2
\]
### Summary
The equation of the circle of radius \( a \) that passes through the points \( (b, 0) \) and \( (-b, 0) \) is:
\[
x^2 + \left(y - \sqrt{a^2 - b^2}\right)^2 = a^2
\]
