Find the equation of the circle which passes through the points (3,4),(3,-6) and (1,2).

Answer

To find the equation of the circle that passes through the points (3, 4), (3, -6), and (1, 2), we can follow these steps:

### Step 1: Write the general equation of a circle
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step 2: Substitute the first point (3, 4)
Substituting the point (3, 4) into the general equation:
\[ (3 - h)^2 + (4 - k)^2 = r^2 \quad \text{(Equation 1)} \]

### Step 3: Substitute the second point (3, -6)
Substituting the point (3, -6) into the general equation:
\[ (3 - h)^2 + (-6 - k)^2 = r^2 \quad \text{(Equation 2)} \]
This can be rewritten as:
\[ (3 - h)^2 + (6 + k)^2 = r^2 \]

### Step 4: Substitute the third point (1, 2)
Substituting the point (1, 2) into the general equation:
\[ (1 - h)^2 + (2 - k)^2 = r^2 \quad \text{(Equation 3)} \]

### Step 5: Set up the equations
Now we have three equations:
1. \((3 - h)^2 + (4 - k)^2 = r^2\)
2. \((3 - h)^2 + (6 + k)^2 = r^2\)
3. \((1 - h)^2 + (2 - k)^2 = r^2\)

### Step 6: Subtract Equation 1 from Equation 2
Subtracting Equation 1 from Equation 2:
\[ (6 + k)^2 - (4 - k)^2 = 0 \]
Expanding both sides:
\[ (36 + 12k + k^2) - (16 - 8k + k^2) = 0 \]
Simplifying:
\[ 36 + 12k + k^2 - 16 + 8k - k^2 = 0 \]
\[ 20 + 20k = 0 \]
This gives:
\[ k = -1 \]

### Step 7: Substitute \(k = -1\) into Equation 1
Now substitute \(k = -1\) into Equation 1:
\[ (3 - h)^2 + (4 + 1)^2 = r^2 \]
\[ (3 - h)^2 + 25 = r^2 \quad \text{(Equation 4)} \]

### Step 8: Subtract Equation 1 from Equation 3
Subtracting Equation 1 from Equation 3:
\[ (1 - h)^2 + (2 - k)^2 - ((3 - h)^2 + (4 - k)^2) = 0 \]
Expanding:
\[ (1 - h)^2 + (2 + 1)^2 - ((3 - h)^2 + (4 + 1)^2) = 0 \]
This simplifies to:
\[ (1 - h)^2 + 9 - ((3 - h)^2 + 25) = 0 \]
Expanding and simplifying:
\[ (1 - h)^2 + 9 - (9 - 6h + h^2 + 25) = 0 \]
\[ 1 - 2h + h^2 + 9 - 9 + 6h - h^2 - 25 = 0 \]
\[ 4h - 25 = 0 \]
Thus:
\[ h = 6.25 \]

### Step 9: Substitute \(h = 6.25\) and \(k = -1\) into Equation 4
Substituting \(h = 6.25\) and \(k = -1\) into Equation 4:
\[ (3 - 6.25)^2 + 25 = r^2 \]
Calculating:
\[ (-3.25)^2 + 25 = r^2 \]
\[ 10.5625 + 25 = r^2 \]
\[ r^2 = 35.5625 \]

### Step 10: Write the final equation of the circle
Now we can write the equation of the circle:
\[ (x - 6.25)^2 + (y + 1)^2 = 35.5625 \]

### Final Answer
The equation of the circle is:
\[ (x - 6.25)^2 + (y + 1)^2 = 35.5625 \]