Find the value of k if the points (1, 3) and (2, k) are conjugated with respect to the circle `x^(2)+y^(2)=35`.

Answer

To find the value of \( k \) such that the points \( (1, 3) \) and \( (2, k) \) are conjugate with respect to the circle defined by the equation \( x^2 + y^2 = 35 \), we can follow these steps:

### Step 1: Understand the condition for conjugate points
For two points \( (x_1, y_1) \) and \( (x_2, y_2) \) to be conjugate with respect to a circle of the form \( x^2 + y^2 = a^2 \), the condition is:
\[ x_1 x_2 + y_1 y_2 = a^2 \]

### Step 2: Identify the points and the value of \( a^2 \)
Here, we have:
- Point 1: \( (x_1, y_1) = (1, 3) \)
- Point 2: \( (x_2, y_2) = (2, k) \)

From the equation of the circle \( x^2 + y^2 = 35 \), we identify \( a^2 = 35 \).

### Step 3: Substitute the points into the conjugate condition
Substituting the values into the conjugate condition:
\[ 1 \cdot 2 + 3 \cdot k = 35 \]

### Step 4: Simplify the equation
This simplifies to:
\[ 2 + 3k = 35 \]

### Step 5: Solve for \( k \)
Now, isolate \( k \):
\[ 3k = 35 - 2 \]
\[ 3k = 33 \]
\[ k = \frac{33}{3} = 11 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{11} \]