A circle has radius 3 units and its centre lies on `y=x-1`. If it passes through the point (7, 3) its equation is ..........

Answer

To find the equation of the circle with a radius of 3 units, centered on the line \(y = x - 1\), and passing through the point (7, 3), we can follow these steps:

### Step 1: Understand the Circle's Equation
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, \(r = 3\), so \(r^2 = 9\).

### Step 2: Determine the Center's Coordinates
Since the center \((h, k)\) lies on the line \(y = x - 1\), we can express \(k\) in terms of \(h\):
\[ k = h - 1 \]
Thus, the center can be represented as \((h, h - 1)\).

### Step 3: Substitute the Center into the Circle's Equation
Substituting \(k = h - 1\) into the circle's equation, we have:
\[ (x - h)^2 + (y - (h - 1))^2 = 9 \]
This simplifies to:
\[ (x - h)^2 + (y - h + 1)^2 = 9 \]

### Step 4: Use the Point (7, 3) to Find \(h\)
Since the circle passes through the point (7, 3), we substitute \(x = 7\) and \(y = 3\) into the equation:
\[ (7 - h)^2 + (3 - (h - 1))^2 = 9 \]
This simplifies to:
\[ (7 - h)^2 + (3 - h + 1)^2 = 9 \]
or
\[ (7 - h)^2 + (4 - h)^2 = 9 \]

### Step 5: Expand and Simplify
Now we expand both squares:
\[ (7 - h)^2 = 49 - 14h + h^2 \]
\[ (4 - h)^2 = 16 - 8h + h^2 \]
Combining these gives:
\[ 49 - 14h + h^2 + 16 - 8h + h^2 = 9 \]
This simplifies to:
\[ 2h^2 - 22h + 65 = 9 \]
Subtracting 9 from both sides:
\[ 2h^2 - 22h + 56 = 0 \]

### Step 6: Solve the Quadratic Equation
Dividing the entire equation by 2:
\[ h^2 - 11h + 28 = 0 \]
Factoring gives:
\[ (h - 4)(h - 7) = 0 \]
Thus, \(h = 4\) or \(h = 7\).

### Step 7: Find Corresponding \(k\) Values
Using \(k = h - 1\):
- If \(h = 4\), then \(k = 3\) (center is \((4, 3)\)).
- If \(h = 7\), then \(k = 6\) (center is \((7, 6)\)).

### Step 8: Write the Equations of the Circles
Now we can write the equations of the two circles:
1. For center \((4, 3)\):
\[ (x - 4)^2 + (y - 3)^2 = 9 \]
2. For center \((7, 6)\):
\[ (x - 7)^2 + (y - 6)^2 = 9 \]

### Final Answer
The equations of the circles are:
1. \((x - 4)^2 + (y - 3)^2 = 9\)
2. \((x - 7)^2 + (y - 6)^2 = 9\)