If (1,2),(3,a) are conjugate points w.r.t `x^(2)+y^(2)=16` then a=

Options

A. 8
B. 7.5
C. 6.5
D. 5.5

Answer

To find the value of \( a \) such that the points \( (1, 2) \) and \( (3, a) \) are conjugate points with respect to the circle given by the equation \( x^2 + y^2 = 16 \), we will use the concept of conjugate points.

### Step-by-step Solution:

1. **Understanding Conjugate Points**:
Conjugate points with respect to a circle satisfy the relationship \( S_1 = 0 \), where \( S_1 \) is defined as:
\[ S_1 = x_1 x + y_1 y - r^2 \]
Here, \( (x_1, y_1) \) is one point and \( (x, y) \) is the other point.

2. **Identifying the Circle**:
The equation of the circle is \( x^2 + y^2 = 16 \). This means the radius \( r = 4 \) (since \( r^2 = 16 \)).

3. **Applying the Formula**:
For the point \( (1, 2) \):
\[ S_1 = 1 \cdot 3 + 2 \cdot a - 16 = 0 \]

4. **Setting Up the Equation**:
Substitute the values into the equation:
\[ 3 + 2a - 16 = 0 \]

5. **Solving for \( a \)**:
Rearranging the equation gives:
\[ 2a - 13 = 0 \]
\[ 2a = 13 \]
\[ a = \frac{13}{2} = 6.5 \]

### Conclusion:
Thus, the value of \( a \) is \( 6.5 \).