If the circles of same radii and with centres (1, 3), (9, 11) cut orthogonally then radius is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To find the radius of the circles that cut orthogonally, we follow these steps:

### Step 1: Understand the condition for orthogonality
The condition for two circles to cut orthogonally is given by:
\[ R_1^2 + R_2^2 = C_1C_2^2 \]
where \( R_1 \) and \( R_2 \) are the radii of the circles, and \( C_1C_2 \) is the distance between the centers of the circles.

### Step 2: Identify the centers of the circles
The centers of the circles are given as:
- Circle 1: \( C_1(1, 3) \)
- Circle 2: \( C_2(9, 11) \)

### Step 3: Calculate the distance between the centers
To find the distance \( C_1C_2 \), we use the distance formula:
\[ C_1C_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ C_1C_2 = \sqrt{(9 - 1)^2 + (11 - 3)^2} = \sqrt{8^2 + 8^2} \]
Calculating this gives:
\[ C_1C_2 = \sqrt{64 + 64} = \sqrt{128} \]

### Step 4: Substitute into the orthogonality condition
Since the circles have the same radius, we denote the radius as \( R \) (i.e., \( R_1 = R_2 = R \)). Thus, the orthogonality condition simplifies to:
\[ 2R^2 = (C_1C_2)^2 \]
Substituting \( C_1C_2 \):
\[ 2R^2 = (\sqrt{128})^2 \]
This simplifies to:
\[ 2R^2 = 128 \]

### Step 5: Solve for the radius \( R \)
Dividing both sides by 2:
\[ R^2 = \frac{128}{2} = 64 \]
Taking the square root of both sides:
\[ R = \sqrt{64} = 8 \]

### Conclusion
The radius of the circles is \( R = 8 \).