Problem 34 - Calculus with Concepts in Calculus

Question

Decide which pairs of lines are parallel, which are perpendicular, and which are neither. For any pair that is not parallel, find the point of intersection. \(x=-1\) and \(y=4\)

Step-by-Step Solution

Answer

The lines are perpendicular, intersecting at (-1, 4).

Step 1: Identify Slope of Line x=-1

The line given by the equation \(x = -1\) is a vertical line. Vertical lines have an undefined slope.

Step 2: Identify Slope of Line y=4

The line given by the equation \(y = 4\) is a horizontal line. Horizontal lines have a slope of 0.

Step 3: Determine Parallelism

Two lines are parallel if they have the same slope. Since \(x = -1\) has an undefined slope and \(y = 4\) has a slope of 0, they do not have the same slope. Thus, they are not parallel.

Step 4: Determine Perpendicularity

Two lines are perpendicular if the product of their slopes is \(-1\). The product of an undefined slope \(x = -1\) and zero slope \(y = 4\) cannot be calculated because one value is undefined. Vertical and horizontal lines are perpendicular by property, since their slopes are effectively opposite; thus, \(x = -1\) and \(y = 4\) are perpendicular.

Step 5: Find Point of Intersection for Non-Parallel Lines

To find the point of intersection for non-parallel lines \(x = -1\) and \(y = 4\), substitute \(x = -1\) into the equation \(y = 4\) and vice versa. The point of intersection is at \((-1, 4)\).

Key Concepts

Parallel Lines

Parallel lines are lines that never meet no matter how far they are extended. This unique feature of parallel lines is due to them sharing the exact same slope.

Imagine two train tracks running side by side. They maintain the same direction as they stretch into the distance and never cross each other. Mathematically speaking, two lines are parallel if their slopes are equal.

• If two lines have the equation forms of, say, \(y = 2x + 1\) and \(y = 2x - 4\), they are parallel because both have a slope of 2.
• Another example could be \(y = -3x + 7\) and \(y = -3x\); both these lines have a slope of -3, making them parallel.

However, in the case of the lines \(x = -1\) and \(y = 4\), they don't share the same slope: one line is vertical, and the other is horizontal, meaning they cannot be parallel. This makes them perpendicular instead, a topic we will dive into next.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). One of the exciting facets of mathematics is how these perpendicular lines behave on a Cartesian plane. When working with slopes, two lines are perpendicular if the product of their slopes is exactly -1.

However, vertical and horizontal lines have a special relationship. Consider the vertical line \(x = -1\) with an undefined slope and the horizontal line \(y = 4\) with a slope of zero.

• Even though the slope product rule cannot be applied here because of the undefined value, we know by property that horizontal and vertical lines are automatically perpendicular.

This perpendicular nature is due to their respective orientations on the Cartesian plane, where one line goes straight up and down and the other straight side to side, meeting at a 90-degree angle.

Point of Intersection

The point of intersection is where two lines meet or cross. For any two non-parallel lines, there is usually one unique point where they intersect.

Identifying this point is like finding where two different paths cross each other. When given the vertical line \(x = -1\) and horizontal line \(y = 4\), both equations are quite straightforward:

• To find the point of intersection, we simply take the values from each line equation and combine them: \(x = -1\) and \(y = 4\).
• This gives us the intersection at the point \((-1, 4)\).

This point is like the crossroad where these lines meet. Understanding how to compute this intersection can be crucial, especially when graphing or working with linear systems. With these basics, unraveling the mysteries of geometry becomes much easier.