Question

For the line \(\mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right), \mathrm{m}\) and \(\mathrm{x}_{1}\) are fixed values, if different lines are drawn according to the different value of \(\mathrm{y}_{1}\), then all such lines would be \(\ldots\) (a) all lines intersect the line \(\mathrm{x}=\mathrm{x}_{1}\) (b) all lines pass through one fixed point (c) all lines are parallel to the line \(\mathrm{y}=\mathrm{x}_{1}\) (d) all lines will be the set of perpendicular lines

Short answer

The correct answer is (a): all lines intersect the line \(x = x_1\).

Step-by-step solution

(Rewriting the given equation in slope-intercept form)

In order to gain insight into the relationship between these lines, we should first rewrite the given equation in slope-intercept form, i.e., \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, we'll solve for \(y\): \(y = mx - mx_1 + y_1\). Notice now that both \(m\) and \(x_1\) are fixed values, and \(y_1\) can vary. This means that the slope of all these lines will be equal to \(m\), but the y-intercept will vary depending on the value of \(y_1\). #Step 2: Analyze the choices given#

(Choosing the correct relationship)

Now, let's examine each choice and see which one is correct based on the equation \(y = mx - mx_1 + y_1\): (a) all lines intersect the line \(x = x_1\): To find this out, we can plug \(x = x_1\) into our equation and see what happens: \(y = m(x_1) - m(x_1) + y_1 \Rightarrow y = y_1\). So, every line intersects the line \(x = x_1\) at the point \((x_1, y_1)\). Thus, this choice is correct. (b) all lines pass through one fixed point: Since the y-intercept varies depending on the value of \(y_1\), the lines do not pass through the same point, and this choice is incorrect. (c) all lines are parallel to the line \(y = x_1\): The line \(y = x_1\) has a slope of 0 and is a horizontal line. The given lines have a fixed slope of \(m\), but no constraint is given on whether \(m\) is 0 or any other value. Therefore, this choice is incorrect, as it cannot be concluded that they are always parallel to the horizontal line \(y = x_1\). (d) all lines will be the set of perpendicular lines: Since all lines have the same fixed slope, \(m\), they cannot be perpendicular to each other, making this choice incorrect. #Conclusion# Based on our analysis, the correct answer is (a): all lines intersect the line \(x = x_1\).

Key concepts

Lines in Coordinate Geometry

Lines in coordinate geometry are a fundamental concept when it comes to understanding the Cartesian plane. These lines are defined by equations that relate the x and y coordinates. You might encounter different forms of line equations:

• **Point-slope form:** \( y - y_1 = m(x - x_1) \)
• **Slope-intercept form:** \( y = mx + b \)
• **Standard form:** \( Ax + By = C \)

Each form has its advantages, and choosing one depends on the problem you need to solve. For instance, the point-slope form is particularly useful when you know a point on the line and its slope, whereas the slope-intercept form easily shows the y-intercept. Understanding these forms is crucial for drawing, analyzing, and interpreting lines on a graph.

Slope of a Line

The slope of a line refers to its steepness and direction. It's calculated as the ratio between the vertical change (rise) and horizontal change (run) between two points on the line. Mathematically, slope \( m \) is expressed as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
A line's slope gives key insights:

• **Positive slope:** The line rises as it moves from left to right.
• **Negative slope:** The line falls as it moves from left to right.
• **Zero slope:** The line is horizontal, showing no vertical change.
• **Undefined slope:** The line is vertical, indicating infinite steepness.

The slope is vital in determining how parallel or perpendicular lines relate to each other. Lines with equal slopes are parallel, while perpendicular lines have slopes that are negative reciprocals.

Line Intersections

Line intersections occur when two lines cross each other at a single point on the coordinate plane. Identifying intersection points involves using the equations of the lines to find their common solution. There are key concepts to consider:

• **Unique intersection:** Two lines with different slopes intersect at one point.
• **Parallel lines:** Lines with the same slope do not intersect.
• **Coincident lines:** Lines with the same slope and intercept coincide, meaning they are the same line.

In the context of the exercise you’re dealing with, analyzing whether options such as parallelism or intersection with a vertical line offer solutions requires transforming line equations into a form you can easily work with, like the slope-intercept form, as done in the given problem.