Problem 11 - The Calculus with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

y = x and y = -x.

Step 1: Identify the Slopes of the Axes

The x-axis has a slope of 0, and the y-axis has an undefined slope. However, we are looking at the angle between these axes.

Step 2: Determine the Angle Bisector in the First Quadrant

The bisector of the angle between the positive x-axis and positive y-axis (first quadrant) will have a slope of 1, as it forms a 45-degree angle with both axes.

Step 3: Determine the Angle Bisector in the Third Quadrant

Similarly, the bisector of the angle between the negative x-axis and negative y-axis (third quadrant) will have a slope of -1.

Step 4: Write the Equations of the Lines

The equations of the lines that pass through the origin with slopes 1 and -1 are: 1. y = x 2. y = -x

Key Concepts

Slope

The slope of a line is a measure of its steepness. It tells us how much the y-coordinate of a point on the line changes for a one-unit change in the x-coordinate. Mathematically, the slope (m) is defined as \( m = \frac{\Delta y}{\Delta x} \) where \( \Delta y \) is the change in the y-coordinate and \( \Delta x \) is the change in the x-coordinate.
For example:

• If the slope is 1, the line rises by 1 unit for each 1 unit it runs to the right.
• If the slope is -1, the line falls by 1 unit for each 1 unit it runs to the right.

In the given exercise, the slopes considered are 1 and -1, which form 45-degree angles with both the x-axis and y-axis in the respective quadrants.

Quadrants

The coordinate plane is divided into four sections called quadrants. They are:

• 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.

In the exercise, we are interested in the first and third quadrants. The line passing through the origin slices the plane into these quadrants at certain angles. For the first quadrant, the angle bisector has a slope of 1, while in the third quadrant, its slope is -1.

Angle Bisector

An angle bisector is a line or ray that divides an angle into two equal parts. For example, in the exercise:

• The angle between the positive x-axis and positive y-axis is 90 degrees. The bisector of this angle in the first quadrant has a 45-degree angle with each axis, resulting in a slope of 1.
• Similarly, the angle between the negative x-axis and negative y-axis in the third quadrant is also 90 degrees. Its bisector has a slope of -1.

The bisectors help us determine the exact equations of the lines that pass through the origin, forming specific angles with the axes.

x-axis and y-axis

The x-axis and y-axis are the two main lines in the Cartesian coordinate plane. They are perpendicular to each other and intersect at the origin (0,0), splitting the plane into four quadrants.
The x-axis is the horizontal line. Any point on it has a y-coordinate of 0.
The y-axis is the vertical line. Any point on it has an x-coordinate of 0.
For this exercise:

• The x-axis (horizontal) has a slope of 0. It extends both positively and negatively in the horizontal direction.
• The y-axis (vertical) has an undefined slope. It extends both positively and negatively in the vertical direction.

Understanding the properties of these axes is crucial in forming equations of lines that interact with them, such as the lines y=x and y=-x in the given problem.