What is the radius of the circle passing through the points (0,0) , (a ,0) and (0,b) ?

Options

A. `sqrt(a^(2) - b^(2))`
B. `sqrt(a^(2) + b^(2))`
C. `(1)/(2) sqrt(a^(2) + b^(2))`
D. `2sqrt(a^(2) + b^(2))`

Answer

To find the radius of the circle passing through the points (0, 0), (a, 0), and (0, b), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**: We have three points:
- Point A: (0, 0)
- Point B: (a, 0)
- Point C: (0, b)

2. **General Equation of Circle**: The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \((-g, -f)\) is the center of the circle.

3. **Substituting Point A (0, 0)**:
- Substitute \(x = 0\) and \(y = 0\) into the equation:
\[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \implies c = 0 \]

4. **Substituting Point B (a, 0)**:
- Substitute \(x = a\) and \(y = 0\):
\[ a^2 + 0^2 + 2g(a) + 0 + 0 = 0 \implies a^2 + 2ga = 0 \]
- Factor out \(a\) (assuming \(a \neq 0\)):
\[ a(a + 2g) = 0 \implies g = -\frac{a}{2} \]

5. **Substituting Point C (0, b)**:
- Substitute \(x = 0\) and \(y = b\):
\[ 0^2 + b^2 + 0 + 2f(b) + 0 = 0 \implies b^2 + 2fb = 0 \]
- Factor out \(b\) (assuming \(b \neq 0\)):
\[ b(b + 2f) = 0 \implies f = -\frac{b}{2} \]

6. **Finding the Center of the Circle**:
- The center of the circle is at \(\left(-g, -f\right)\):
\[ \left(-\left(-\frac{a}{2}\right), -\left(-\frac{b}{2}\right)\right) = \left(\frac{a}{2}, \frac{b}{2}\right) \]

7. **Calculating the Radius**:
- The radius \(r\) can be found using the distance formula from the center to any of the points, say (0, 0):
\[ r = \sqrt{\left(\frac{a}{2} - 0\right)^2 + \left(\frac{b}{2} - 0\right)^2} \]
- Simplifying this:
\[ r = \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{b}{2}\right)^2} = \sqrt{\frac{a^2}{4} + \frac{b^2}{4}} = \sqrt{\frac{a^2 + b^2}{4}} = \frac{1}{2}\sqrt{a^2 + b^2} \]

### Final Answer:
The radius of the circle passing through the points (0, 0), (a, 0), and (0, b) is:
\[ r = \frac{1}{2} \sqrt{a^2 + b^2} \]