Question

Consider the equation $y - y_1 = m(x - x_1)$. If $m$ and $x_1$ 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 lines intersect the line x=x1
• D. all lines will be parallel to the line y=x1

Answer 1

The Correct Option is B, C

We are given the equation of a line in point-slope form: \[ y - y_1 = m(x - x_1) \] where \( m \) and \( x_1 \) are fixed, and different lines are drawn by varying \( y_1 \).

Step 1: General form of the line
Rearranging the equation: \[ y = m(x - x_1) + y_1 \] This shows that the value of \( y_1 \) shifts the line vertically for each different value of \( y_1 \), while the slope \( m \) and \( x_1 \) remain fixed. 

Step 2: Analysis of the lines
Since \( m \) is fixed, all lines will have the same slope and therefore will be parallel to each other. 

Step 3: Intersection with the line \( x = x_1 \)
If we substitute \( x = x_1 \) into the equation of the line: \[ y = m(x_1 - x_1) + y_1 = y_1 \] Thus, each line will intersect the vertical line \( x = x_1 \) at a point \( (x_1, y_1) \), where \( y_1 \) is the specific value for each line.

Answer:

\[ \boxed{\text{There will be a set of parallel lines and all lines intersect the line } x = x_1.} \]