Find the equation of the circle through the points (0,0), (2,0) and (0,4). Also find the coordinates of its centre and its radius.

Answer

To find the equation of the circle that passes through the points (0,0), (2,0), and (0,4), we will follow these steps:

### Step 1: Write the general equation of a circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \((h, k)\) is the center of the circle, \(g = -h\), \(f = -k\), and \(c\) is a constant.

### Step 2: Substitute the first point (0,0)
Substituting the point (0,0) into the equation:
\[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \]
This simplifies to:
\[ c = 0 \]

### Step 3: Substitute the second point (2,0)
Now substitute the point (2,0):
\[ 2^2 + 0^2 + 2g(2) + 2f(0) + c = 0 \]
This simplifies to:
\[ 4 + 4g + 0 + 0 = 0 \]
Thus, we have:
\[ 4g = -4 \implies g = -1 \]

### Step 4: Substitute the third point (0,4)
Now substitute the point (0,4):
\[ 0^2 + 4^2 + 2g(0) + 2f(4) + c = 0 \]
This simplifies to:
\[ 16 + 0 + 8f + 0 = 0 \]
Thus, we have:
\[ 8f = -16 \implies f = -2 \]

### Step 5: Write the equation of the circle
Now we have \(g = -1\), \(f = -2\), and \(c = 0\). Substituting these values back into the general equation:
\[ x^2 + y^2 + 2(-1)x + 2(-2)y + 0 = 0 \]
This simplifies to:
\[ x^2 + y^2 - 2x - 4y = 0 \]

### Step 6: Find the center of the circle
The center of the circle can be found using the formulas:
\[ \text{Center} = (-g, -f) = (1, 2) \]

### Step 7: Find the radius of the circle
The radius \(r\) can be calculated using the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]
Substituting the values:
\[ r = \sqrt{(-1)^2 + (-2)^2 - 0} = \sqrt{1 + 4} = \sqrt{5} \]

### Conclusion
Thus, the equation of the circle is:
\[ x^2 + y^2 - 2x - 4y = 0 \]
The coordinates of its center are \((1, 2)\) and the radius is \(\sqrt{5}\) units.

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