Problem 48 - Calculus with Concepts in Calculus

Question

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: y+4=-\frac{3}{5}\left(x-\frac{1}{2}\right) ; P=\left(-1, \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{5}{3}x + \frac{13}{6} \).

Step 1: Identify the Slope of Given Line

The equation of the given line is in point-slope form: \( y + 4 = -\frac{3}{5}(x - \frac{1}{2}) \). From this equation, we can see that the slope \( m_1 \) of the line \( l \) is \( -\frac{3}{5} \).

Step 2: Find the Slope of the Perpendicular Line

Since perpendicular lines have slopes that are negative reciprocals of each other, the slope \( m_2 \) of the line we want to find is the negative reciprocal of \( -\frac{3}{5} \). Calculating this gives \( m_2 = \frac{5}{3} \).

Step 3: Write the Equation of the Perpendicular Line in Point-Slope Form

Using the point \( P(-1, \frac{1}{2}) \) and the slope \( \frac{5}{3} \), plug these into the point-slope form: \( y - y_1 = m(x - x_1) \). This results in: \( y - \frac{1}{2} = \frac{5}{3}(x + 1) \).

Step 4: Simplify the Equation

Expand and simplify the equation. Distribute \( \frac{5}{3} \) over \( (x + 1) \) to get \( y - \frac{1}{2} = \frac{5}{3}x + \frac{5}{3} \). Then add \( \frac{1}{2} \) to both sides to isolate \( y \). The simplified equation becomes \( y = \frac{5}{3}x + \frac{13}{6} \).

Key Concepts

Point-Slope Form

The point-slope form is a very useful way to write the equation of a line. It's mostly used when you know:

• a point on the line,
• and the slope of the line.

The formula for point-slope form is:\[ y - y_1 = m(x - x_1) \]Here:

• \( (x_1, y_1) \) is a point on the line.
• \( m \) is the slope of the line.

To use point-slope form, you just need to plug in the values you know. This is what we did in the solution when we found the equation for the line. We knew the slope and a point, so we used those in the formula. This method is quick and shows you how the line looks based on just one point and the slope.

Using point-slope form makes finding the equation very straightforward. Once you know a single point of where your line crosses and the direction or steepness of the line, you're all set to write the equation.

Negative Reciprocal

The concept of a negative reciprocal is important when dealing with perpendicular lines. If you know the slope of one line, you can easily find the slope of a line perpendicular to it. Here's how it works:

• First, understand what a reciprocal is: It's when you flip the fraction. For example, the reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \).
• The negative reciprocal changes the sign as well. So, starting with \( -\frac{3}{5} \), the negative reciprocal is \( \frac{5}{3} \).

Think of it like this: if one line has a gentle downhill slope, a line perpendicular will have a steep uphill slope, because you flip and change the direction. Finding the negative reciprocal is simple:
1. Flip the fraction.2. Change the sign.

In the problem given, the line \( l \) had a slope of \( -\frac{3}{5} \). We flipped the fraction to get \( \frac{5}{3} \) and changed its sign to find the perpendicular slope. By knowing the slope of the given line, you're equipped to find the slope of its perpendicular counterpart.

Slope of a Line

The slope of a line measures its steepness and direction. It can be seen in how much y changes for a change in x between two points on the line. The formula for finding the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]A positive slope indicates a line rising from left to right, while a negative slope shows a line falling from left to right. A zero slope means a horizontal line, and an undefined slope refers to a vertical line (where \( x_1 = x_2 \)).

• If the line goes upwards as you move along it from left to right, it has a positive slope.
• If it goes downwards, it has a negative slope.
• If it is flat and horizontal, the slope is zero.
• If it is vertical, the slope is undefined.

Knowing the slope helps understand the line's behavior. In our problem, recognizing that the original line had a slope of \( -\frac{3}{5} \) helped us determine that the line perpendicular to it must have a slope that is the negative reciprocal, which was \( \frac{5}{3} \). This idea is central in geometric problems involving angles and inclinations.