Problem 42 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-2,5)\) and perpendicular to the vertical line passing through \((1,4)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(0x + 1y = 5\).

Step 1: Understand the Requirements

We need to find the equation of a line that is perpendicular to a given vertical line and passes through a specific point. The given point is (-2, 5). The given vertical line passes through (1, 4), so the line is x=1.

Step 2: Determine the Slope of the Required Line

Vertical lines have an undefined slope. Hence, a line perpendicular to a vertical line is horizontal. Horizontal lines have a slope of 0.

Step 3: Use Point-Slope Form

We use the point-slope formula: \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1) = (-2, 5)\) and \(m = 0\). Substituting these values, the equation becomes \(y - 5 = 0(x + 2)\), which simplifies to \(y = 5\).

Step 4: Convert to Standard Form

The standard form of a line is \(Ax + By = C\). For \(y = 5\), we rewrite it as \(0 \cdot x + 1 \cdot y = 5\). Therefore, the standard form of the line is \(0x + 1y = 5\).

Key Concepts

Standard Form

When we talk about lines in math, the "standard form" of a line is a very common way to present linear equations. The standard form is written as \(Ax + By = C\). This means the equation is structured with all the variables on one side and the constant on the other. The values \(A\), \(B\), and \(C\) are integers, and \(A\) should not be negative (though different scenarios may have different conventions).
One great thing about the standard form is that it's easy to see important things, like whether two lines are parallel or when they might intersect.

• **Parallel lines** in standard form have the same \(A/B\) ratio.
• It's straightforward to convert equations from some other forms into standard form, making it adaptable for various calculations.

Using standard form can help quickly organize and understand the properties of a line or a group of lines.

Perpendicular Lines

Perpendicular lines can be a bit tricky at first, but the concept is truly fascinating once you get the hang of it. In geometry, two lines are said to be perpendicular if they intersect at a right angle (90 degrees).
If two lines are perpendicular in a two-dimensional Cartesian plane, their slopes multiply to \ -1.

• For example, if one line has a slope of \(3\), a line perpendicular to it will have a slope of \(-\frac{1}{3}\).
• However, when dealing with the unique cases of vertical and horizontal lines, a vertical line (which has an undefined slope) is always perpendicular to any horizontal line (slope of 0).

In situations like the exercise where a line must be perpendicular to a vertical line, it becomes horizontal with a straightforward equation like \(y = 5\). This is because the horizontal line with zero slope is the natural perpendicular counterpart to the undefined slope of a vertical line.

Point-Slope Form

One of the most intuitive ways to write the equation of a line is using the point-slope form formula: \ \(y - y_1 = m(x - x_1)\).
This formula is fantastic because it allows you to easily create an equation of a line when you know:

• A single point the line passes through \((x_1, y_1)\) and
• The slope \(m\) of the line.

Let's break this down using the exercise: The line we're interested in passes through the point \((-2, 5)\) and is horizontal, meaning \(m = 0\).
Plugging in these values into the formula, you get: \ \(y - 5 = 0(x + 2)\) which simply becomes \(y = 5\).
The simplicity of the point-slope form makes it super easy to handle problems, regardless of whether you're tackling straightforward or complex line equations.