Problem 58 - Precalculus with Limits

Question

In Exercises 51-58, find the distance between the point and the line. \(\textit{Point}\) \((4, 2)\) \(\textit{Line}\) \(x - y = 20\)

Step-by-Step Solution

Answer

The shortest distance between the point (4,2) and the line \(x - y = 20\) is \(9\sqrt{2}\) units.

Step 1: Arranging the Line Equation in Standard Form

The given line equation is \(x - y = 20\). In the standard form, it becomes \(x - y - 20 = 0\). Here, A=1, B=-1, and C=-20.

Step 2: Identifying the Point Coordinates

The given point is (4,2). Thus, x1=4 and y1=2.

Step 3: Substituting into the Formula

Substitute the values of A, B, C, x1, y1 into the equation. You get \( \frac{|1*4 - 1*2 - 20|}{\sqrt{1^2 + (-1)^2}} \) which simplifies to \( \frac{|-18|}{\sqrt{2}} \) and finally results as \( \frac{18\sqrt{2}}{2} \) which simplifies to \(9\sqrt{2}\).

Key Concepts

Standard Form of a Line Equation

Understanding the standard form of a line equation is crucial for solving many algebraic and geometric problems. It is expressed as Ax + By + C = 0, where A, B, and C are constants.

The beauty of this form is that it neatly categorizes all the characteristics of the line. Coefficient A corresponds to the x-component, B to the y-component, and C represents the line's intercept with the y-axis when x is zero. If both A and B are not zero, the line is neither horizontal nor vertical.

In our exercise, the equation x - y = 20 is converted to standard form by subtracting 20 from both sides, resulting in x - y - 20 = 0. Here, A equals 1, B equals -1, and C is -20. This standardized form allows us to use algebraic methods to find various properties of the line, such as the distance from a point to the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the position of points, lines, and shapes in a two-dimensional space. The fundamental elements are the x and y axes, which perpendicularly intersect at the origin (0,0), creating a grid where every point can be located by a pair of numbers called coordinates.

Each coordinate pair (x, y) represents a unique position in the plane. For example, the point (4, 2) in our exercise indicates a position that is four units to the right and two units up from the origin. By knowing these coordinates, we can solve problems and find relationships between geometric shapes and equations, such as calculating the distance of a point from a line, as seen in the given exercise.

Perpendicular Distance Formula

When it comes to finding the shortest distance between a point and a line, which is always the perpendicular distance, we can use the perpendicular distance formula. This formula is defined as D = |Ax1 + By1 + C| / √(A² + B²), where (x1, y1) are the coordinates of the point, and A, B, and C are the coefficients from the standard form of the line equation.

This distance is a measure of how far away the point is from the line along a line that's perpendicular to it, which is the shortest possible distance between the two. In our exercise, plugging in the values into the formula gives us D = |1*4 - 1*2 - 20| / √(1² + (-1)²), simplifying to |-18| / √2, which equals 9√2. It's important to note that we use absolute value to ensure the distance is a non-negative value. Utilizing this formula is a powerful method for solving geometry problems involving distances.