If (6,8),(k,2) are inverse points w.r.t the circle `x^(2)+y^(2)=25` then 2k=

Options

A. 1
B. 3
C. 5
D. 7

Answer

To solve the problem, we need to find the value of \(2k\) given that the points \((6,8)\) and \((k,2)\) are inverse points with respect to the circle defined by the equation \(x^2 + y^2 = 25\).

### Step-by-step Solution:

1. **Understand the concept of inverse points**:
For two points \((x_1, y_1)\) and \((x_2, y_2)\) to be inverse points with respect to a circle defined by \(x^2 + y^2 = r^2\), the following relationship must hold:
\[ x_1 \cdot x_2 + y_1 \cdot y_2 = r^2 \]
Here, \(r^2 = 25\), so \(r = 5\).

2. **Assign the points**:
Let \((x_1, y_1) = (6, 8)\) and \((x_2, y_2) = (k, 2)\).

3. **Set up the equation**:
Using the inverse point relationship:
\[ 6k + 8 \cdot 2 = 25 \]

4. **Simplify the equation**:
Calculate \(8 \cdot 2\):
\[ 6k + 16 = 25 \]

5. **Isolate \(k\)**:
Subtract 16 from both sides:
\[ 6k = 25 - 16 \]
\[ 6k = 9 \]

6. **Solve for \(k\)**:
Divide both sides by 6:
\[ k = \frac{9}{6} = \frac{3}{2} \]

7. **Find \(2k\)**:
Multiply \(k\) by 2:
\[ 2k = 2 \cdot \frac{3}{2} = 3 \]

### Final Answer:
Thus, the value of \(2k\) is \(3\).