Find the equation of the circle which passes through the points (0,0), (0,1) and (2,3)

Answer

To find the equation of the circle that passes through the points (0,0), (0,1), and (2,3), we can follow these steps:

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

### Step 2: Substitute the first point (0,0)
Substituting the point (0,0) into the equation:
\[ (0 - h)^2 + (0 - k)^2 = r^2 \]
This simplifies to:
\[ h^2 + k^2 = r^2 \quad \text{(Equation 1)} \]

### Step 3: Substitute the second point (0,1)
Now, substitute the point (0,1):
\[ (0 - h)^2 + (1 - k)^2 = r^2 \]
This simplifies to:
\[ h^2 + (1 - k)^2 = r^2 \]
Expanding \((1 - k)^2\):
\[ h^2 + (1 - 2k + k^2) = r^2 \]
Thus, we have:
\[ h^2 + 1 - 2k + k^2 = r^2 \quad \text{(Equation 2)} \]

### Step 4: Substitute the third point (2,3)
Now, substitute the point (2,3):
\[ (2 - h)^2 + (3 - k)^2 = r^2 \]
This expands to:
\[ (2 - h)^2 + (3 - k)^2 = r^2 \]
Expanding both squares:
\[ (4 - 4h + h^2) + (9 - 6k + k^2) = r^2 \]
So we have:
\[ h^2 - 4h + k^2 - 6k + 13 = r^2 \quad \text{(Equation 3)} \]

### Step 5: Set up equations to eliminate \(r^2\)
From Equation 1, we know \(r^2 = h^2 + k^2\). We can substitute \(r^2\) from Equation 1 into Equations 2 and 3.

#### Substitute into Equation 2:
\[ h^2 + 1 - 2k + k^2 = h^2 + k^2 \]
This simplifies to:
\[ 1 - 2k = 0 \implies k = \frac{1}{2} \]

#### Substitute into Equation 3:
\[ h^2 - 4h + k^2 - 6k + 13 = h^2 + k^2 \]
Substituting \(k = \frac{1}{2}\):
\[ h^2 - 4h + \left(\frac{1}{2}\right)^2 - 6\left(\frac{1}{2}\right) + 13 = h^2 + \left(\frac{1}{2}\right)^2 \]
This simplifies to:
\[ h^2 - 4h + \frac{1}{4} - 3 + 13 = h^2 + \frac{1}{4} \]
Cancelling \(h^2\) and \(\frac{1}{4}\) from both sides gives:
\[ -4h + 10 = 0 \implies 4h = 10 \implies h = \frac{5}{2} \]

### Step 6: Find \(r^2\)
Now substitute \(h\) and \(k\) back into Equation 1 to find \(r^2\):
\[ r^2 = h^2 + k^2 = \left(\frac{5}{2}\right)^2 + \left(\frac{1}{2}\right)^2 \]
Calculating:
\[ r^2 = \frac{25}{4} + \frac{1}{4} = \frac{26}{4} = \frac{13}{2} \]

### Step 7: Write the final equation of the circle
Now we have \(h = \frac{5}{2}\), \(k = \frac{1}{2}\), and \(r^2 = \frac{13}{2}\). The equation of the circle is:
\[ \left(x - \frac{5}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 = \frac{13}{2} \]

### Final Answer
The required equation of the circle is:
\[ \left(x - \frac{5}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 = \frac{13}{2} \]