Two circles `C _(1) and C _(2)` with equal radii r are centered at `(3,2) and (6,5)` respectively. If `C _(1) and C _(2)` intersect orthogonally, then r is equal to.

Answer

To solve the problem of finding the radius \( r \) of two circles \( C_1 \) and \( C_2 \) that intersect orthogonally, we can follow these steps:

### Step 1: Identify the centers of the circles
The centers of the circles are given as:
- Center of circle \( C_1 \) (denoted as \( K_1 \)): \( (3, 2) \)
- Center of circle \( C_2 \) (denoted as \( K_2 \)): \( (6, 5) \)

### Step 2: Calculate the distance between the centers
To find the distance \( d \) between the centers \( K_1 \) and \( K_2 \), we use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(6 - 3)^2 + (5 - 2)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 3: Apply the condition for orthogonal intersection
For two circles to intersect orthogonally, the following condition must hold:
\[ r_1^2 + r_2^2 = d^2 \]
Since both circles have equal radii \( r \), we can write:
\[ r^2 + r^2 = d^2 \implies 2r^2 = d^2 \]

### Step 4: Substitute the distance into the equation
From Step 2, we found that \( d = 3\sqrt{2} \). Therefore, we can substitute this into our equation:
\[ 2r^2 = (3\sqrt{2})^2 \]
Calculating the right side:
\[ (3\sqrt{2})^2 = 9 \times 2 = 18 \]
So we have:
\[ 2r^2 = 18 \]

### Step 5: Solve for \( r^2 \)
Now, divide both sides by 2:
\[ r^2 = \frac{18}{2} = 9 \]

### Step 6: Find \( r \)
Taking the square root of both sides gives:
\[ r = \sqrt{9} = 3 \]

Thus, the radius \( r \) of the circles is \( 3 \) units.

### Final Answer:
The radius \( r \) is equal to \( 3 \).

---