Equation of the circle with centre `((a)/(2), (b)/(2))` and radius `sqrt ((a ^(2) + b ^(2))/(4 ))` is

Options

A. `x ^(2) +y ^(2) -ax - by = (a + b) ^(2)`
B. `x ^(2) + y ^(2) + ax + by = (a -b) ^(2) `
C. `x ^(2) + y ^(2) -ax -by = ((a ^(2) + b ^(2)))/( 4) `
D. `x ^(2) + y ^(2) -ax - by =0`

Answer

To find the equation of the circle with the given center and radius, we follow these steps:

### Step 1: Identify the center and radius
The center of the circle is given as \((\frac{a}{2}, \frac{b}{2})\) and the radius is given as \(\sqrt{\frac{a^2 + b^2}{4}}\).

### Step 2: Use the standard equation of a circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, \(h = \frac{a}{2}\), \(k = \frac{b}{2}\), and \(r = \sqrt{\frac{a^2 + b^2}{4}}\).

### Step 3: Substitute the values into the equation
Substituting \(h\), \(k\), and \(r\) into the standard equation:
\[ \left(x - \frac{a}{2}\right)^2 + \left(y - \frac{b}{2}\right)^2 = \left(\sqrt{\frac{a^2 + b^2}{4}}\right)^2 \]

### Step 4: Simplify the right side
Calculating the right side:
\[ \left(\sqrt{\frac{a^2 + b^2}{4}}\right)^2 = \frac{a^2 + b^2}{4} \]

### Step 5: Write the equation
Now, we have:
\[ \left(x - \frac{a}{2}\right)^2 + \left(y - \frac{b}{2}\right)^2 = \frac{a^2 + b^2}{4} \]

### Step 6: Expand the left side
Expanding the left side:
\[ \left(x - \frac{a}{2}\right)^2 = x^2 - ax + \frac{a^2}{4} \]
\[ \left(y - \frac{b}{2}\right)^2 = y^2 - by + \frac{b^2}{4} \]
Combining these, we get:
\[ x^2 - ax + \frac{a^2}{4} + y^2 - by + \frac{b^2}{4} = \frac{a^2 + b^2}{4} \]

### Step 7: Combine terms
Combining all terms gives:
\[ x^2 + y^2 - ax - by + \frac{a^2}{4} + \frac{b^2}{4} = \frac{a^2 + b^2}{4} \]

### Step 8: Move constant to the right side
Subtract \(\frac{a^2 + b^2}{4}\) from both sides:
\[ x^2 + y^2 - ax - by = 0 \]

### Final Equation
Thus, the equation of the circle is:
\[ x^2 + y^2 - ax - by = 0 \]