Radius of the circle `(x - a) (x - b) +(y - p) (y - q)=0` is .........

Answer

To find the radius of the circle given by the equation \((x - a)(x - b) + (y - p)(y - q) = 0\), we can follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:
\[ (x - a)(x - b) + (y - p)(y - q) = 0 \]

### Step 2: Identify the endpoints of the diameter
From the equation, we can identify the endpoints of the diameter of the circle as \( (a, p) \) and \( (b, q) \).

### Step 3: Calculate the distance of the diameter
The distance \(D\) between the two points \((a, p)\) and \((b, q)\) can be calculated using the distance formula:
\[ D = \sqrt{(b - a)^2 + (q - p)^2} \]

### Step 4: Find the radius
The radius \(r\) of the circle is half of the diameter:
\[ r = \frac{D}{2} = \frac{1}{2} \sqrt{(b - a)^2 + (q - p)^2} \]

### Final Answer
Thus, the radius of the circle is:
\[ r = \frac{1}{2} \sqrt{(b - a)^2 + (q - p)^2} \]

---