If the line joining the points `(_x_1, y_1) and (x_2, y_2)` subtends a right angle at the point (1,1), then `x_1 + x_2 + y_2 + y_2` is equal to

Options

A. 4
B. 5
C. 6
D. 8

Answer

To solve the problem, we need to find the value of \( x_1 + x_2 + y_1 + y_2 \) given that the line joining the points \( (x_1, y_1) \) and \( (x_2, y_2) \) subtends a right angle at the point \( (1, 1) \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
We have three points: \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(1, 1) \). The line segment \( AB \) subtends a right angle at point \( C \).

2. **Using the Slope Condition**:
For two lines to be perpendicular, the product of their slopes must equal \(-1\). We will find the slopes of the segments \( AC \) and \( BC \).

- The slope of line \( AC \) is given by:
\[ \text{slope of } AC = \frac{y_1 - 1}{x_1 - 1} \]
- The slope of line \( BC \) is given by:
\[ \text{slope of } BC = \frac{y_2 - 1}{x_2 - 1} \]

3. **Setting Up the Equation**:
Since \( AC \) is perpendicular to \( BC \), we have:
\[ \frac{y_1 - 1}{x_1 - 1} \cdot \frac{y_2 - 1}{x_2 - 1} = -1 \]

4. **Cross-Multiplying**:
Rearranging the equation gives:
\[ (y_1 - 1)(y_2 - 1) = -(x_1 - 1)(x_2 - 1) \]

5. **Expanding Both Sides**:
Expanding both sides, we get:
\[ y_1y_2 - y_1 - y_2 + 1 = - (x_1x_2 - x_1 - x_2 + 1) \]
This simplifies to:
\[ y_1y_2 - y_1 - y_2 + 1 = -x_1x_2 + x_1 + x_2 - 1 \]

6. **Rearranging the Equation**:
Bringing all terms to one side gives:
\[ y_1y_2 + x_1x_2 - y_1 - y_2 + x_1 + x_2 + 2 = 0 \]

7. **Finding the Required Sum**:
We need to find \( x_1 + x_2 + y_1 + y_2 \). Rearranging the equation gives:
\[ x_1 + x_2 + y_1 + y_2 = y_1y_2 + x_1x_2 + 2 \]

8. **Conclusion**:
Since we have established the relationship, we can conclude that:
\[ x_1 + x_2 + y_1 + y_2 = 2 \]

### Final Answer:
The value of \( x_1 + x_2 + y_1 + y_2 \) is \( 2 \).

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