Consider the equation `y-y_1=m(x-x_1)` . If `ma n dx_1` are fixed and different lines are drawn for different values of `y_1,` then (a) the lines will pass through a fixed point (b) there will be a set of parallel lines (c) all the lines intersect the line `x=x_1` (d)all the lines will be parallel to the line `y=x_1`

Options

A. the lines will pass through a fixed point
B. there will be a set of parallel lines
C. all the lines intersect the line `x=x_(1)`
D. all the lines will be parallel to the line `y=x_(1)`

Explanation

To solve the problem, we start with the equation of the line given by:

\[ y - y_1 = m(x - x_1) \]

where \( m \) and \( x_1 \) are fixed values, and \( y_1 \) can vary. We need to analyze the implications of this equation as \( y_1 \) changes.

### Step-by-Step Solution:

1. **Identify the Fixed and Variable Components:**
- Here, \( m \) (the slope) and \( x_1 \) (the x-coordinate of a fixed point) are constants.
- \( y_1 \) is the variable that will change, leading to different lines.

2. **Rearranging the Equation:**
- We can rearrange the equation to express \( y \) in terms of \( x \) and \( y_1 \):
\[ y = m(x - x_1) + y_1 \]
- This shows that for each fixed \( x_1 \) and \( m \), the value of \( y \) depends on \( y_1 \).

3. **Understanding the Lines:**
- Since \( m \) is fixed, the slope of the lines remains constant.
- As \( y_1 \) varies, the y-intercept of the line changes, but the slope \( m \) does not.

4. **Conclusion about the Lines:**
- Because the slope is constant (fixed \( m \)), all lines drawn for different values of \( y_1 \) will be parallel to each other.
- Therefore, the set of lines will be parallel lines.

5. **Intersection with the Line \( x = x_1 \):**
- All these lines will intersect the vertical line \( x = x_1 \) at the point where \( x = x_1 \) and \( y = y_1 \).
- Thus, for any value of \( y_1 \), the intersection point will always lie on the line \( x = x_1 \).

### Final Answers:
- (b) There will be a set of parallel lines.
- (c) All the lines intersect the line \( x = x_1 \).