Centre of circle , passing through (0,0) ,(a,0) and (0,b) , is

Options

A. `((b)/(2),(a)/(2))`
B. `((a)/(2),(b)/(2))`
C. (b,a)
D. (a,b)

Answer

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

### Step 1: Write the general equation of the circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \((-g, -f)\) represents the center of the circle.

### 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 (a, 0)
Now substitute the point (a, 0) into the equation:
\[ a^2 + 0^2 + 2g(a) + 2f(0) + c = 0 \]
Since \(c = 0\), we have:
\[ a^2 + 2ga = 0 \]
Factoring out \(a\):
\[ a(a + 2g) = 0 \]
Since \(a \neq 0\), we can set:
\[ a + 2g = 0 \implies 2g = -a \implies g = -\frac{a}{2} \]

### Step 4: Substitute the third point (0, b)
Now substitute the point (0, b) into the equation:
\[ 0^2 + b^2 + 2g(0) + 2f(b) + c = 0 \]
Again, since \(c = 0\), we have:
\[ b^2 + 2fb = 0 \]
Factoring out \(b\):
\[ b(b + 2f) = 0 \]
Since \(b \neq 0\), we can set:
\[ b + 2f = 0 \implies 2f = -b \implies f = -\frac{b}{2} \]

### Step 5: Find the center of the circle
Now that we have the values of \(g\) and \(f\):
\[ g = -\frac{a}{2}, \quad f = -\frac{b}{2} \]
The center of the circle is given by:
\[ (-g, -f) = \left(-\left(-\frac{a}{2}\right), -\left(-\frac{b}{2}\right)\right) = \left(\frac{a}{2}, \frac{b}{2}\right) \]

### Final Answer
The center of the circle passing through the points (0, 0), (a, 0), and (0, b) is:
\[ \left(\frac{a}{2}, \frac{b}{2}\right) \]