Problem 19 - Calculus

Question

Use the facts that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals of one another. Find equations for the lines through the point \((a, b)\) that are parallel and perpendicular to the line \(y=m x+c\) assuming \(m \neq 0\)

Step-by-Step Solution

Answer

Parallel: \( y = mx + (b - ma) \); Perpendicular: \( y = -\frac{1}{m}x + (b + \frac{a}{m}) \).

Step 1: Identify Slope of Given Line

The given line is in the form \( y = mx + c \). From this equation, we can identify the slope \( m \).

Step 2: Determine Slope of Parallel Line

A line parallel to another line shares the same slope. Therefore, the slope of the line through \((a, b)\) parallel to \( y = mx + c \) is also \( m \).

Step 3: Write Equation for Parallel Line

Using the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \((x_1, y_1) = (a, b)\), the equation of the parallel line is \( y - b = m(x - a) \). Simplifying, we get the equation for the parallel line: \( y = mx + (b - ma) \).

Step 4: Determine Slope of Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the other line's slope. Hence, the slope of the perpendicular line is \( -\frac{1}{m} \).

Step 5: Write Equation for Perpendicular Line

Using the point-slope form again for the perpendicular line through \((a, b)\), we have \( y - b = -\frac{1}{m}(x - a) \). Simplifying, the equation becomes \( y = -\frac{1}{m}x + (b + \frac{a}{m}) \).

Key Concepts

Parallel Lines

Lines in a plane that never intersect, no matter how far they are extended, are called parallel lines. A key feature of parallel lines is that they have the same slope. The slope is a measure of how steep a line is. If two lines have the same slope, they will run in the same direction and never meet.

In our problem, we begin with a line given in slope-intercept form:

• Equation: \( y = mx + c \), where \( m \) is the slope.

Given a point \((a, b)\) through which our parallel line must pass, we will use this slope \( m \) because a line parallel to another retains the slope. We use the point-slope form, which helps us incorporate the given point, to determine the equation of the parallel line:

• Point-slope form: \( y - y_1 = m(x - x_1) \)
• Where the point \((x_1, y_1) = (a, b) \)
• The equation simplifies to: \( y = mx + (b - ma) \).

This formula helps us construct any line parallel to another, ensuring they both share the same direction.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). The unique characteristic of perpendicular lines is that their slopes are negative reciprocals of each other. This means that when the slope of one line (\(m\)) is flipped and its sign is changed, we obtain the slope of the line perpendicular to it.

Let's look at how this applies in our exercise:

• Given line: \( y = mx + c \)
• Slope of the perpendicular line: \(-\frac{1}{m} \)

Once we know the perpendicular slope, we can find the equation of the line passing through point \((a, b)\) using the point-slope form.

The point-slope form of the equation becomes:

• \( y - b = -\frac{1}{m}(x - a) \)
• Simplifying yields: \( y = -\frac{1}{m}x + (b + \frac{a}{m}) \)

This approach lets us create a line that not only goes through a desired point but also maintains that specific perpendicular angle with the given line.

Slope-Intercept Form

The slope-intercept form of a line is one of the most commonly used methods for writing equations of lines. Its format is straightforward and helps to identify the slope and the y-intercept at a glance.

The slope-intercept form is given by:

• \( y = mx + c \)
• Where \( m \) represents the slope
• \( c \) is the y-intercept, the point where the line crosses the y-axis

This form is especially useful when analyzing lines:

• Easy identification of slope \( m \)
• Quick determination of the y-intercept \(c\)

Using the slope-intercept form is practical in both constructing new lines and understanding relationships such as parallelism and perpendicularity between different lines. By evaluating these components, it allows you to graph lines quickly and accurately, displaying both direction and placement on the Cartesian plane.