If (1, a), (b, 2) are conjugate points with renpcet to the circle `x^(2)+y^(2)=25`, then 4a+2b=

Options

A. 25
B. 50
C. 100
D. 150

Answer

To solve the problem, we need to find the value of \(4a + 2b\) given that the points \((1, a)\) and \((b, 2)\) are conjugate points with respect to the circle defined by the equation \(x^2 + y^2 = 25\).

### Step-by-step Solution:

1. **Understand the Circle Equation**: The equation of the circle is given as \(x^2 + y^2 = 25\). This means the center of the circle is at the origin (0, 0) and the radius is 5 (since \(\sqrt{25} = 5\)).

2. **Identify the Points**: We have two points: \((1, a)\) and \((b, 2)\).

3. **Use the Conjugate Points Condition**: For points \((x_1, y_1)\) and \((x_2, y_2)\) to be conjugate with respect to a circle defined by \(x^2 + y^2 = r^2\), the following condition must hold:
\[ x_1 x_2 + y_1 y_2 = r^2 \]
Here, \(r^2 = 25\).

4. **Substitute the Points into the Condition**:
- Let \((x_1, y_1) = (1, a)\) and \((x_2, y_2) = (b, 2)\).
- Substitute into the conjugate condition:
\[ 1 \cdot b + a \cdot 2 = 25 \]
This simplifies to:
\[ b + 2a = 25 \quad \text{(Equation 1)} \]

5. **Rearranging the Equation**: We can rearrange Equation 1 to express \(b\) in terms of \(a\):
\[ b = 25 - 2a \quad \text{(Equation 2)} \]

6. **Find \(4a + 2b\)**: We need to find \(4a + 2b\). Substitute Equation 2 into this expression:
\[ 4a + 2b = 4a + 2(25 - 2a) \]
Simplifying this gives:
\[ 4a + 50 - 4a = 50 \]

7. **Final Result**: Therefore, the value of \(4a + 2b\) is:
\[ \boxed{50} \]