Problem 71 - Calculus Single Variable

Question

The distance of a point \(P\) to a line \(\ell\) is the distance of \(P\) to \(Q\) where \(Q\) is the point on \(\ell\) such that the line segment \(\overline{P Q}\) is perpendicular to \(\ell\). By elementary geometry \(|\overline{P Q}|<\left|\overline{P Q}^{\prime}\right|\) if \(Q^{\prime}\) is any other point on \(\ell .\) Prove that if \(A x+B y+C=0\) is a general linear equation for \(\ell\) and \(P=\left(x_{0}, y_{0}\right),\) then $$ \frac{\left|A x_{0}+B y_{0}+C\right|}{\sqrt{A^{2}+B^{2}}} $$ is the distance from \(P\) to \(\ell\).

Step-by-Step Solution

Answer

The distance is \(\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\).

Step 1: Understand the Point-to-Line Distance Formula

The formula for the distance from a point \(P(x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by \(\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\). We'll prove this using the geometric definition of distance as the length of the perpendicular from the point to the line.

Step 2: Define the Perpendicular Line

To find the perpendicular distance, we need to find where the perpendicular line that passes through \(P(x_0, y_0)\) intersects the line \(Ax + By + C = 0\). The slope of the line \(y = mx + b\) is \(-\frac{A}{B}\), so the slope of the perpendicular line is \(\frac{B}{A}\). Therefore, the equation of the line perpendicular and passing through \(P(x_0, y_0)\) is \(y - y_0 = \frac{B}{A}(x - x_0)\).

Step 3: Find the Intersection Point Q

To find the intersection point \(Q(x_1, y_1)\), solve the system of equations for the two lines: \(Ax + By + C = 0\) and \(y - y_0 = \frac{B}{A}(x - x_0)\). Substitute \(y = y_0 + \frac{B}{A}(x - x_0)\) into the line equation to find \(x_1\), then solve for \(y_1\).

Step 4: Calculate the Distance between P and Q

Using the coordinates \((x_1, y_1)\) from the previous step, calculate the distance \(PQ\) using the distance formula: \(\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}\).

Step 5: Derive the Point-to-line Distance Formula

Substitute \(x_1\) and \(y_1\) into the distance formula, showing simplifications along the way. Verify that after simplification, the distance \(PQ\) is equal to \(\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\).

Key Concepts

Point-to-line distance

In geometry, calculating the distance from a point to a line is a common problem. This distance is defined as the shortest route from the point to any point on the line. Imagine this as a straight path, like the sunbeam shining from a spotlight to your face—it takes the shortest path. For a point \(P(x_0, y_0)\) and a line defined by the equation \(Ax + By + C = 0\), the distance can be determined with the formula:

\[ \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

This formula is powerful as it provides a direct method without needing to find the specific perpendicular point \(Q\) on the line. It's useful in various applications, like physics, engineering, and anywhere you need to calculate precise straight-line distances.

Linear equations

At the heart of this distance problem is the concept of linear equations. A linear equation in two variables, such as \(Ax + By + C = 0\), represents a straight line on a Cartesian coordinate plane. The coefficients \(A\) and \(B\) determine the slope of the line, while \(C\) affects its position.

• The slope of the line is \(-\frac{A}{B}\), influencing its steepness and direction.
• Linear equations are the foundational elements of algebra and calculus, modeling various real-world situations.

Understanding how to manipulate these equations is crucial as they form the basis for finding intersections and perpendiculars, which are key to solving distance problems.

Geometric proof

The idea of proving concepts using geometry is fundamental. In proving the point-to-line distance formula, a geometric method involves constructing a perpendicular from the point to the line and showing this is the shortest distance.

Key Steps in Geometric Proof

• Identify the equations of the given line and the line perpendicular to it passing through the point of interest.
• Calculate where these lines intersect—this point is \(Q\).
• Compute the distance \(|PQ|\) using the Pythagorean theorem, which is the basis for the distance formula.

These steps not only verify the formula mathematically but also provide a visual understanding of the relationship between points and lines.

Distance formula

In various mathematical analyses, calculating distance between points is crucial. The distance formula is derived from the Pythagorean theorem, used widely in geometry.

Understanding the Distance Formula

For two points \((x_0, y_0)\) and \((x_1, y_1)\), the Euclidean distance \(|PQ|\) is computed as:

\[ \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2} \]

• It calculates the straight-line distance "as the crow flies."
• Plays a crucial role in determining the shortest path in physical and theoretical spaces.

When applied to points on a Cartesian plane, it remains indispensable in calculations throughout mathematics and related scientific fields.