Problem 10 - Calculus for Biology and Medicine

Question

How close does the line \(y=1+2 x\) come to the origin?

Step-by-Step Solution

Answer

The shortest distance from the line to the origin is \(\frac{1}{\sqrt{5}}\).

Step 1: Understand the problem

We need to find the shortest distance from the origin (0,0) to the line given by the equation \(y = 1 + 2x\). This involves using the formula for the distance from a point to a line.

Step 2: Recall the distance formula

The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by \[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}.\] Here, the line equation \(y = 1 + 2x\) can be rewritten as \(2x - y + 1 = 0\). So, \(A = 2\), \(B = -1\), and \(C = 1\).

Step 3: Plug the point and line parameters into the formula

Using the distance formula with the origin \((0, 0)\) as the point, we have: \[ d = \frac{|2(0) - 1(0) + 1|}{\sqrt{2^2 + (-1)^2}} = \frac{|1|}{\sqrt{4 + 1}}.\]

Step 4: Calculate the distance

Simplifying further, we find: \[ d = \frac{1}{\sqrt{5}}.\] This represents the shortest distance from the origin to the line.

Key Concepts

Distance Formula

One of the fundamental concepts in geometry is determining the shortest distance between a point and a line. The distance formula serves this purpose well. It provides a method to calculate the shortest distance from a specific point \((x_0, y_0)\) to any line expressed in the standard form \(Ax + By + C = 0\). The formula is:

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

The components of the formula include:

• **Numerator**: \(Ax_0 + By_0 + C\) is the algebraic expression calculated by substituting the point coordinates into the line equation.
• **Denominator**: \(\sqrt{A^2 + B^2}\) represents the length of the vector normal to the line.

For our specific problem, the line \(y = 1 + 2x\) is rewritten as \(2x - y + 1 = 0\), identifying \(A = 2\), \(B = -1\), and \(C = 1\). Calculating the distance to the origin \( (0, 0) \), we substitute into the formula and arrive at a distance of \({1}{\sqrt{5}}\), which is mathematically simplified for ease of understanding.

Analytical Geometry

Analytical geometry, sometimes referred to as coordinate geometry, is an essential branch of mathematics that uses algebraic symbolism to describe and solve geometric problems. This way of envisioning geometrical shapes through algebra marks a shift from classical geometric concepts.At its core, analytical geometry uses coordinates and equations to describe:

• The positions of points,
• The properties and relationships of lines, and
• The geometric shapes formed by such relationships.

In the exercise, the line is represented by the equation \(y = 1 + 2x\), which translates graphically into a straight line. By rewriting it into \(2x - y + 1 = 0\), we put it in the standard form suitable for using various analytical methods, such as the distance formula. This capability to swiftly switch between visual graphs and algebraic expressions is the hallmark of analytical geometry.

Equations of Lines

The equation of a line is a fundamental component of analytical geometry, representing a linear relationship between two variables. There are several forms of line equations, with the slope-intercept form \(y = mx + b\) being one of the most intuitive.The slope-intercept form allows us to:

• Understand the slope \(m\), indicating the steepness or incline of the line, and
• Determine the y-intercept \(b\), which is the point where the line crosses the y-axis.

In our problem, the line \(y = 1 + 2x\) shows a slope of \(2\) and a y-intercept of \(1\). Converting this into the general form \(2x - y + 1 = 0\) aids us in applying formulas like the distance formula efficiently. Understanding these different representations helps us solve problems related to lines and angles, ensuring a robust grasp of geometry through algebra.