If (1,4), (-2,3) are conjugate points w.r.t `x^(2)+y^(2)=k` then k=

Options

A. 10
B. `sqrt(10)`
C. 100
D. 4

Answer

To find the value of \( k \) such that the points \( (1, 4) \) and \( (-2, 3) \) are conjugate points with respect to the circle defined by the equation \( x^2 + y^2 = k \), 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 points with respect to the circle \( x^2 + y^2 = k \), the following condition must hold:
\[ x_1 x_2 + y_1 y_2 = k \]

### Step 2: Assign values to the points
Let \( (x_1, y_1) = (1, 4) \) and \( (x_2, y_2) = (-2, 3) \).

### Step 3: Substitute the values into the condition
Now, substitute the values into the conjugate point condition:
\[ 1 \cdot (-2) + 4 \cdot 3 = k \]

### Step 4: Calculate the left-hand side
Calculate the left-hand side:
\[ -2 + 12 = k \]

### Step 5: Simplify to find \( k \)
Now, simplify the expression:
\[ k = 10 \]

### Conclusion
Thus, the value of \( k \) is \( 10 \).