Problem 55 - Precalculus with Limits

Question

In Exercises 51-58, find the distance between the point and the line. \(\textit{Point}\) \((6, 2)\) \(\textit{Line}\) \(x + 1 = 0\)

Step-by-Step Solution

Answer

The shortest distance from the point (6,2) to the line x = -1 is 7 units.

Step 1: Identify the Given Point and Line

The point is given as (6,2) and the line's equation is given as x + 1 = 0 which can also be written as x = -1. This equation represents a vertical line parallel to the y-axis at x = -1.

Step 2: Find the Distance using the Formula

For the vertical line x = a, the distance (d) from the point (x1, y1) is simply \(d = |x1 - a|\). Here, \(x1 = 6\), \(y1 = 2\) and \(a = -1\). Implementing these values into the formula we get \(d = |-1 - 6| = |-7| = 7\).

Step 3: Writing the final answer

This implies that the shortest distance from the point (6,2) to the line x = -1 is 7 units.

Key Concepts

Vertical Line

A vertical line is one of the most straightforward types of lines in the coordinate plane. It runs straight up and down and is parallel to the y-axis. In mathematical terms, a vertical line can be represented by an equation of the form \(x = a\), where \(a\) is a constant.
- The key feature of a vertical line is that every point on the line shares the same x-coordinate.- No matter what y-value you pick, the x-value remains constant.In the context of our exercise, the line is described by the equation \(x + 1 = 0\), which simplifies to \(x = -1\). This means our vertical line passes through every point with an x-coordinate of -1. Vertical lines are important in mathematics because they help us understand parallelism and distances in the coordinate plane.

Distance Formula

Finding distances in geometry often involves using specific formulas. When dealing with a point and a vertical line, the distance formula becomes quite simple.
For vertical lines, the formula to calculate the distance from a point \((x_1, y_1)\) to a line \(x = a\) is:\[d = |x_1 - a|\]- Here, \(d\) stands for distance.- \(x_1\) is the x-coordinate of the point.- \(a\) is the x-coordinate where the vertical line exists.The absolute value keeps the distance non-negative, which makes sense because distance is always positive or zero. In our example, applying this formula helps us find that the distance between point \((6, 2)\) and the line \(x = -1\) is 7 units. Understanding these elements ensures you can solve similar problems effortlessly in the future.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and shapes.
- It is defined by two perpendicular number lines intersecting at the origin (0,0).- The horizontal axis is called the x-axis, and the vertical one is the y-axis.Every location on this plane is identified by a pair of coordinates \((x, y)\). This systematic approach helps us to precisely determine locations and measure distances, like the distance between our point (6, 2) and the vertical line \(x = -1\). Understanding the coordinate plane lets you visualize the relationship between mathematical concepts and the real world. It serves as a foundation for exploring geometry and algebra in more depth.

Absolute Value

Absolute value is an essential concept in mathematics that looks at the size of a number, ignoring its sign. It is always non-negative.
- Represented by two vertical bars, like \(|-7|\).- \(|x|\) is the distance from zero to \(x\) on the number line.In the context of calculating distance, especially in our exercise, absolute value simplifies finding how far a point is from a line. It assures that we measure distance in positive terms. For example, \(|-7| = 7\) confirms that the distance from the point \((6, 2)\) to the line \(x = -1\) is indeed 7 units. By understanding absolute value, you can confidently solve problems involving distance and many other areas of mathematics.