Problem 13 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { Through the origin and bisecting the angle between the axes in the second and fourth quadrants. } $$

Step-by-Step Solution

Answer

y = x or y = -x.

Step 1 - Understand the problem

The line goes through the origin and bisects the angle between the x-axis and y-axis. We need to determine its equation through these conditions.

Step 2 - Determine the angle bisectors

In the coordinate plane, the angle between the positive x-axis and y-axis is 90 degrees. The bisectors of this angle will form lines at 45-degree angles with the axes. For the second quadrant, the angle bisector will have a slope of 1 (forming a 135-degree angle with the positive x-axis), and for the fourth quadrant, the angle bisector will have a slope of -1 (forming a 315-degree angle with the positive x-axis).

Step 3 - Form equations using slopes

Using the slope-intercept form of a line equation, which is given by y = mx + c ^symbol, where m is the slope and c is the y-intercept. Since the line passes through the origin (0,0), the y-intercept c = 0. Hence, the equations of the lines will be: y = x and y = -x

Step 4 - Verify conditions

Both slopes form lines which bisect the angle between the x-axis and y-axis in the second and fourth quadrants respectively.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most common ways to describe the equation of a line. It is written as \( y = mx + c \) where:

• \( y \) is the dependent variable
• \( x \) is the independent variable
• \( m \) is the slope of the line
• \( c \) is the y-intercept of the line, the point where the line crosses the y-axis

Understanding this form is crucial because it allows you to easily determine the slope and the starting point of the line. For example, if you have a slope of 2 and a y-intercept of 3, your equation would be \( y = 2x + 3 \).In this exercise, since the line passes through the origin (0,0), the y-intercept \( c = 0 \), simplifying the equation to \( y = mx \). This form makes it easier to understand and solve.

angle bisectors

An angle bisector is a line or ray that divides an angle into two congruent angles. In the coordinate plane, the x-axis and y-axis intersect at a 90-degree angle. If a line bisects this angle, it splits the 90-degree angle into two 45-degree angles. For this exercise, the challenge is to find the equations for the lines that bisect the angle between the x-axis and y-axis in the second and fourth quadrants.
The angle in the second quadrant with slopes 1 and -1 creates an angle bisector of 45 degrees. Therefore, lines with a slope m of 1 or -1 will bisect the angle in their respective quadrants.

• The line with a slope = 1 is represented by \( y = x \) which lies in the second quadrant.
• The line with a slope = -1 is represented by \( y = -x \) which lies in the fourth quadrant.

Both of these lines indeed divide the angle formed by the x-axis and y-axis evenly, fulfilling the angle bisector condition.

coordinate plane

A coordinate plane, also known as the Cartesian plane, is a two-dimensional plane formed by the intersection of a vertical line called the y-axis and a horizontal line called the x-axis. These axes divide the plane into four quadrants:

• First Quadrant: Both x and y are positive
• Second Quadrant: x is negative, y is positive
• Third Quadrant: Both x and y are negative
• Fourth Quadrant: x is positive, y is negative

When working with problems in the coordinate plane, understanding the position of each quadrant and the behavior of linear equations in these quadrants is important.
In the given exercise, the goal is to find lines passing through the origin and bisecting the angles between the axes in the second and fourth quadrants. For the second quadrant, the angle bisector slope is 1, leading to the equation \( y = x \). For the fourth quadrant, the angle bisector slope is -1, leading to the equation \( y = -x \).