If a straight line passes through `(x_(1),y_(1))` and its segment between the axes is bisected at this point then its equation is given by

Options

A. `x/(x_(1))+y/(y_(1))=2`
B. `2(xy_(1)+yx_(1))=x_(1)y_(1)`
C. `xy_(1)+yx_(1)=x_(1)y_(1)`
D. none

Answer

To find the equation of a straight line that passes through the point \((x_1, y_1)\) and has its segment between the axes bisected at this point, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find the equation of a line that passes through the point \((x_1, y_1)\).
- The segment of the line between the x-axis and y-axis is bisected at this point.

2. **Identify the Intercepts**:
- Let the x-intercept be \(a\) (where the line intersects the x-axis, so the point is \((a, 0)\)).
- Let the y-intercept be \(b\) (where the line intersects the y-axis, so the point is \((0, b)\)).

3. **Using the Midpoint Formula**:
- The midpoint \(P\) of the segment joining the points \((a, 0)\) and \((0, b)\) is given by:
\[ P = \left(\frac{a + 0}{2}, \frac{0 + b}{2}\right) = \left(\frac{a}{2}, \frac{b}{2}\right) \]
- Since the midpoint \(P\) is also the point \((x_1, y_1)\), we can equate:
\[ \frac{a}{2} = x_1 \quad \text{and} \quad \frac{b}{2} = y_1 \]

4. **Solving for \(a\) and \(b\)**:
- From \(\frac{a}{2} = x_1\), we find:
\[ a = 2x_1 \]
- From \(\frac{b}{2} = y_1\), we find:
\[ b = 2y_1 \]

5. **Equation of the Line in Intercept Form**:
- The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
- Substituting the values of \(a\) and \(b\):
\[ \frac{x}{2x_1} + \frac{y}{2y_1} = 1 \]

6. **Simplifying the Equation**:
- Multiply through by 2 to eliminate the denominators:
\[ \frac{x}{x_1} + \frac{y}{y_1} = 2 \]

7. **Final Equation**:
- Thus, the equation of the line that passes through the point \((x_1, y_1)\) and has its segment between the axes bisected at this point is:
\[ \frac{x}{x_1} + \frac{y}{y_1} = 2 \]