The length of the common chord of two circles of radii `r_1` and `r_2` which intersect at right angles is

Options

A. `(r_1r_2)/sqrt(r_1^2+r_2^2)`
B. `(2r_1r_2)/sqrt(r_1^2+r_2^2)`
C. `(r_1+r_2)/sqrt(r_1^2+r_2^2)`
D. `(r_1r_2)/(r_1^2+r_2^2)`

Answer

To find the length of the common chord of two circles with radii \( r_1 \) and \( r_2 \) that intersect at right angles, we can follow these steps:

### Step 1: Understand the Geometry
When two circles intersect at right angles, the angle between their radii at the point of intersection is \( 90^\circ \). This means that the radius of one circle is perpendicular to the radius of the other circle at the point of intersection.

### Step 2: Use the Right Triangle
Let the centers of the circles be \( O_1 \) and \( O_2 \), and let the points of intersection be \( A \) and \( B \). The line segment \( O_1O_2 \) forms the hypotenuse of a right triangle \( O_1ABO_2 \).

### Step 3: Apply the Pythagorean Theorem
In triangle \( O_1O_2A \):
\[ O_1A^2 + O_2A^2 = O_1O_2^2 \]
Where:
- \( O_1A = r_1 \) (radius of the first circle)
- \( O_2A = r_2 \) (radius of the second circle)

Thus, we have:
\[ r_1^2 + r_2^2 = O_1O_2^2 \]

### Step 4: Length of the Common Chord
The length of the common chord \( AB \) can be derived from the relationship involving the area of triangles formed. The area of triangle \( O_1AB \) can be expressed in two ways:
1. Using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times O_1A \times O_2A \times \sin(90^\circ) = \frac{1}{2} \times r_1 \times r_2 \]
2. Using the height from \( O_1O_2 \) to the chord \( AB \):
\[ \text{Area} = \frac{1}{2} \times h \times AB \]
where \( h \) is the distance from the center line \( O_1O_2 \) to the chord \( AB \).

### Step 5: Relate the Two Areas
Equating the two expressions for the area:
\[ \frac{1}{2} r_1 r_2 = \frac{1}{2} h \times AB \]
This simplifies to:
\[ r_1 r_2 = h \times AB \]

### Step 6: Find the Height \( h \)
From the right triangle formed, we can find \( h \) using:
\[ h = \sqrt{r_1^2 + r_2^2} \]

### Step 7: Substitute \( h \) Back
Substituting \( h \) back into the area equation:
\[ r_1 r_2 = \sqrt{r_1^2 + r_2^2} \times AB \]
Solving for \( AB \):
\[ AB = \frac{r_1 r_2}{\sqrt{r_1^2 + r_2^2}} \]

### Final Result
Thus, the length of the common chord of the two circles is:
\[ AB = \frac{2 r_1 r_2}{\sqrt{r_1^2 + r_2^2}} \]

### Conclusion
The correct answer is:
\[ \text{Length of the common chord} = \frac{2 r_1 r_2}{\sqrt{r_1^2 + r_2^2}} \]