Problem 65 - Precalculus

Question

Find the distance between the point and the line. $$\begin{array}{cc}\text{Point} && \text{Line} \\ (1,-3) && y=2 x-5\end{array}$$

Step-by-Step Solution

Answer

The distance between the point (1,-3) and the line \(y=2x-5\) is 0 units.

Step 1: Identifying the Given Parameters

The given point is (1,-3). The line's equation is in the form of \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, the slope \(m = 2\) and y-intercept \(c = -5\) in the equation \(y=2x-5\).

Step 2: Applying the Perpendicular Distance Formula from a Point to a Line

The formula for finding the perpendicular distance from a point \((x1, y1 )\) to a line \((y= mx +c)\) is: \[ D = \frac{|m . x_1 - y_1 + c|}{\sqrt{m^2 + 1}} \] Substitute the values of \(x_1 = 1\), \(y_1 = -3\), \(m = 2\) and \(c = -5\), into the formula.

Step 3: Simplifying the Expression

By substituting the values into the formula, the final expression to be evaluated is: \[ D = \frac{|2 . 1 - (-3) -5 |}{\sqrt{2^2 + 1}} = \frac{2 + 3 - 5}{\sqrt{4 + 1}} = \frac{0}{\sqrt{5}} = 0\]

Key Concepts

Perpendicular Distance Formula

Understanding how to calculate the perpendicular distance from a point to a line is essential in coordinate geometry. The formula seems complex at first, but it's designed to give the shortest distance between any point and a line in a two-dimensional space.

Here's the general formula for calculating this distance:\[ D = \frac{|m \cdot x_1 - y_1 + c|}{\sqrt{m^2 + 1}} \] To make sense of it, let's break it down:

• \(D\) represents the distance we want to find.
• \(m\) is the slope of the line.
• \(x_1, y_1\) are the coordinates of our point.
• \(c\) is the y-intercept of the line.

The formula calculates the absolute value of the line equation applied to the point's x-coordinate and subtracts the y-coordinate, adjusts for the y-intercept, and then divides by the square root of the square of the slope plus one. This might seem technical, but what's happening geometrically is we're finding the length of the segment perpendicular to the line that connects to our point, hence the 'perpendicular distance'.

The value inside the absolute value sign can be thought of as 'how far off' the y-coordinate of our point is from the y-coordinate of the line at that particular x. The square root of \(m^2 + 1\) is associated with the Pythagorean theorem, as we are dealing with right triangles when talking about perpendicular distances.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method enables problems in geometry to be solved using algebra by translating geometric shapes into algebraic equations. To solve a problem like finding the distance from a point to a line, the following are important:

• The Cartesian Coordinate System allows us to locate points using ordered pairs \(x, y\))
• Each point has a unique position defined by these pairs.
• Lines can also be defined by equations, which explain the relationship between x and y coordinates of every point on the line.

With these principles, coordinate geometry allows us to precisely calculate distances and other geometrical relationships on a plane.

Line Equation

In coordinate geometry, lines are often described using equations, enabling a clear relationship between all the points along the line. The most common form of the line equation is the slope-intercept form, expressed as \(y = mx + c\). In this equation:

• \(y\) and \(x\) continue to represent coordinates on the line.
• \(m\) is the slope of the line, indicating its steepness and direction.
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

This form of the line equation is straightforward to use for finding a line's slope and intercept, which makes it particularly useful for solving various geometric problems, such as calculating the distance from a point to the line.