The co-ordinates of A and B are `(x_(1), y_(1))` and `(x_(2), y_(2))` and O is the origin. If circles be described on OA, OB as diameters, then length of common chord is

Options

A. `(x_(1)y_(2)-x_(2)y_(1))//AB`
B. `(x_(1)y_(1)-x_(2)y_(2))//AB`
C. `(x_(1)y_(2)+x_(2)y_(1))//AB`
D. `(x_(1)y_(1)+x_(2)y_(2))//AB`

Answer

To find the length of the common chord of the circles described on OA and OB as diameters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points:**
- Let the coordinates of point A be \( (x_1, y_1) \).
- Let the coordinates of point B be \( (x_2, y_2) \).
- The origin O has coordinates \( (0, 0) \).

2. **Understanding the Circles:**
- Circle 1 is described on OA as a diameter.
- Circle 2 is described on OB as a diameter.
- The center of Circle 1 is at the midpoint of OA, and the center of Circle 2 is at the midpoint of OB.

3. **Finding the Area of Triangle ABO:**
- The area of triangle ABO can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1y_2 - x_2y_1 \right| \]

4. **Using the Length of Chord:**
- Let OC be the length of the common chord we want to find, and AB be the length of the line segment joining points A and B.
- The area of triangle ABO can also be expressed in terms of OC and AB:
\[ \text{Area} = \frac{1}{2} \times OC \times AB \]

5. **Setting the Areas Equal:**
- Since both expressions represent the area of triangle ABO, we can set them equal to each other:
\[ \frac{1}{2} \left| x_1y_2 - x_2y_1 \right| = \frac{1}{2} \times OC \times AB \]

6. **Solving for OC:**
- Cancel the \( \frac{1}{2} \) from both sides:
\[ \left| x_1y_2 - x_2y_1 \right| = OC \times AB \]
- Rearranging gives:
\[ OC = \frac{\left| x_1y_2 - x_2y_1 \right|}{AB} \]

7. **Finding the Length of AB:**
- The length of AB can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

8. **Final Expression for OC:**
- Substitute \( AB \) back into the equation for \( OC \):
\[ OC = \frac{\left| x_1y_2 - x_2y_1 \right|}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}} \]

### Final Answer:
The length of the common chord is given by:
\[ \text{Length of common chord} = \frac{x_1y_2 - x_2y_1}{AB} \]