If the circles of same radii and with centres (2,3),(5,6) cut orthogonally then radius is

Options

A. 3
B. 5
C. 6
D. 2

Answer

To solve the problem, we need to find the radius of two circles that cut orthogonally and have the same radius. The centers of the circles are given as (2, 3) and (5, 6).

### Step-by-Step Solution:

1. **Identify the Centers and Radii**:
- Let the centers of the circles be \( C_1(2, 3) \) and \( C_2(5, 6) \).
- Since both circles have the same radius, we denote the radius as \( R \).

2. **Use the Orthogonality Condition**:
- The circles cut orthogonally if the sum of the squares of their radii equals the square of the distance between their centers.
- Mathematically, this is expressed as:
\[ R_1^2 + R_2^2 = d^2 \]
- Since \( R_1 = R_2 = R \), we can write:
\[ 2R^2 = d^2 \]

3. **Calculate the Distance Between the Centers**:
- The distance \( d \) between the centers \( C_1(2, 3) \) and \( C_2(5, 6) \) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Substituting the coordinates:
\[ d = \sqrt{(5 - 2)^2 + (6 - 3)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \]

4. **Set Up the Equation**:
- Now substituting \( d^2 \) into the orthogonality condition:
\[ 2R^2 = (\sqrt{18})^2 \]
- This simplifies to:
\[ 2R^2 = 18 \]

5. **Solve for R**:
- Dividing both sides by 2:
\[ R^2 = 9 \]
- Taking the square root of both sides gives:
\[ R = 3 \]

6. **Conclusion**:
- Therefore, the radius of the circles is \( R = 3 \).