Problem 34 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Through \((2,6) ; \quad\) perpendicular to the line \(y=1\)

Step-by-Step Solution

Answer

The equation is \(x = 2\).

Step 1: Understand the Given Line

The given line is defined by the equation \( y = 1 \), which is a horizontal line. Horizontal lines have a slope of 0.

Step 2: Determine the Slope of the Perpendicular Line

A line perpendicular to a horizontal line has an undefined slope. This corresponds to a vertical line.

Step 3: Find the Equation of the Perpendicular Line

Since vertical lines have an undefined slope and pass through a specific x-coordinate, the equation of the line passing through \((2, 6)\) and perpendicular to \(y = 1\) is \(x = 2\).

Key Concepts

Perpendicular Lines

Perpendicular lines are two lines that intersect to form a right angle, which is exactly 90 degrees. When it comes to their slopes, perpendicular lines have a special relationship. If you have the slope of one line, the slope of a line perpendicular to it can be found by taking the negative reciprocal. This means you flip the fraction and change its sign. For example, if one line has a slope of \(m\), the perpendicular line will have a slope of \(-\frac{1}{m}\).
But there is an exception! If a line is perfectly horizontal or vertical, its perpendicular counterpart won’t have a slope in the traditional sense. A horizontal line, for example, runs parallel to the x-axis, and a perpendicular line to it would be vertical.
It's important to remember this distinction because it helps in identifying the relationship between perpendicular lines and in finding their equations.

Horizontal and Vertical Lines

Horizontal and vertical lines form the backbone of coordinate geometries, offering unique properties. A horizontal line runs left to right and is parallel to the x-axis. Its equation is always in the form of \(y = c\), where \(c\) is the y-coordinate of any point on the line. The slope of a horizontal line is zero, meaning there’s no vertical change as you move along the line.
In contrast, a vertical line moves up and down, parallel to the y-axis. Its equation is in the form \(x = k\), where \(k\) is the x-coordinate of any point on the line. The slope of a vertical line is undefined because it involves division by zero, symbolizing an infinite or abrupt change that cannot be quantified.
Understanding these two types of lines is crucial since they are frequent in different exercises, especially in scenarios involving perpendicular lines.

Slope of a Line

The slope of a line measures its steepness, indicating how much the line rises or falls as it moves horizontally. It is calculated as the ratio of vertical change (rise) to horizontal change (run) between two points on a line, expressed as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This ratio helps in identifying how lines behave and in predicting where they'll move next.
The slope can be:

• Positive: the line ascends from left to right.
• Negative: the line descends from left to right.
• Zero: the line is horizontal.
• Undefined: the line is vertical.

Grasping the concept of slope is vital in solving equations of lines because it impacts both the visual representation and mathematical equations of the line. Recognizing these effects ensures that you accurately identify relationships between different lines, especially when dealing with perpendicular or parallel line scenarios.