Find the equation of the circle which passes through the points `(7, 0)`, (0, sqrt(7))` and has its centre on the x-axis.

Explanation

To find the equation of the circle that passes through the points (7, 0) and (0, √7), and has its center on the x-axis, we can follow these steps:

### Step 1: Define the center of the circle
Since the center of the circle lies on the x-axis, we can denote the center as \( (h, 0) \), where \( h \) is the x-coordinate of the center.

### Step 2: Use the distance formula
The radius of the circle is the distance from the center to any point on the circle. We can use the distance formula to set up equations based on the two points through which the circle passes.

1. **Distance from center to point (7, 0)**:
\[ OP = \sqrt{(h - 7)^2 + (0 - 0)^2} = |h - 7| \]

2. **Distance from center to point (0, √7)**:
\[ OQ = \sqrt{(h - 0)^2 + (0 - \sqrt{7})^2} = \sqrt{h^2 + 7} \]

### Step 3: Set the distances equal
Since both distances represent the radius of the circle, we can set them equal to each other:
\[ |h - 7| = \sqrt{h^2 + 7} \]

### Step 4: Solve the equation
We will consider two cases for the absolute value.

#### Case 1: \( h - 7 = \sqrt{h^2 + 7} \)
1. Square both sides:
\[ (h - 7)^2 = h^2 + 7 \]
\[ h^2 - 14h + 49 = h^2 + 7 \]
\[ -14h + 49 - 7 = 0 \]
\[ -14h + 42 = 0 \implies h = 3 \]

#### Case 2: \( h - 7 = -\sqrt{h^2 + 7} \)
1. Square both sides:
\[ (h - 7)^2 = (h^2 + 7) \]
\[ h^2 - 14h + 49 = h^2 + 7 \]
\[ -14h + 49 - 7 = 0 \]
\[ -14h + 42 = 0 \implies h = 3 \]

In both cases, we find that \( h = 3 \).

### Step 5: Find the radius
Now that we have the center of the circle as \( (3, 0) \), we can find the radius by calculating the distance from the center to one of the points, say (7, 0):
\[ \text{Radius} = OP = |3 - 7| = 4 \]

### Step 6: Write the equation of the circle
The standard form of the equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 3 \), \( k = 0 \), and \( r = 4 \):
\[ (x - 3)^2 + (y - 0)^2 = 4^2 \]
\[ (x - 3)^2 + y^2 = 16 \]

### Step 7: Expand the equation
Expanding the equation:
\[ (x^2 - 6x + 9) + y^2 = 16 \]
\[ x^2 + y^2 - 6x + 9 - 16 = 0 \]
\[ x^2 + y^2 - 6x - 7 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 6x - 7 = 0 \]