Problem 82 - College Algebra and Calculus: An Applied Approach

Question

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(4,8),(-4,2) ; L_{2}:(3,-5),\left(-1, \frac{1}{3}\right)\)

Step-by-Step Solution

Answer

After doing the calculations in the steps, we can conclude whether the lines are parallel, perpendicular, or neither.

Step 1: Calculate the slope of the first line

The slope of line \(L_{1}\) that passes through points \((4,8)\) and \((-4,2)\) can be calculated using the formula: \[ m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \]where:\( (x_{1}, y_{1}) = (4, 8) \)and:\( (x_{2}, y_{2}) = (-4, 2) \)Substitute these values into the slope formula to find the slope of line \(L_{1}\).

Step 2: Calculate the slope of the second line

The slope of line \(L_{2}\) that passes through points \((3, -5)\) and \(\left(-1, \frac{1}{3}\right)\) can be calculated using the same formula. Here:\( (x_{1}, y_{1}) = (3, -5) \)and\( (x_{2}, y_{2}) = \left(-1, \frac{1}{3}\right) \)Substitute these values into the slope formula to find the slope of line \(L_{2}\).

Step 3: Compare the slopes

Compare the slopes of \(L_{1}\) and \(L_{2}\) calculated in the first two steps. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. If neither is true, then the lines are neither parallel nor perpendicular.

Key Concepts

Slope Calculation

The concept of slope in coordinate geometry helps us understand the steepness and direction of a line on a plane. The slope formula is given by the expression: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where points \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two points on the line. This formula essentially captures the change in the y-values compared to the change in the x-values over the line.
- **Positive Slope**: If the slope is positive, it means the line is ascending as it goes from left to right.
- **Negative Slope**: Conversely, a negative slope indicates that the line descends from left to right.
- **Zero Slope**: A zero slope implies a horizontal line, indicating consistent y-values across x-values.
- **Undefined Slope**: When the slope is undefined, it typically represents a vertical line where the x-values remain constant.In the context of the exercise, calculating the slopes of lines \( L_1 \) and \( L_2 \) involves plugging the coordinates of their respective points into the slope formula. This process allows us to numerically determine how each line moves across the Cartesian plane.

Parallel and Perpendicular Lines

Understanding parallel and perpendicular lines involves examining their slopes.
**Parallel Lines**
Two lines are considered parallel if they share the same slope, meaning \( m_1 = m_2 \). For lines that are parallel, they will never intersect, no matter how far they are extended on the coordinate plane. Given this nature, parallel lines always remain equidistant from each other.

**Perpendicular Lines**
Perpendicular lines intersect at a 90-degree angle, and the slopes of these lines are negative reciprocals of each other. This means that if \( m_1 \) is the slope of one line, the slope of a perpendicular line would be \( m_2 = -\frac{1}{m_1} \). Whenever two lines are perpendicular, they form right angles at their intersection points.In the exercise provided, comparing the slopes calculated for \( L_1 \) and \( L_2 \) helps determine whether these lines are parallel, perpendicular, or neither, by checking if their slopes are equal or negative reciprocals.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to describe the properties and relations of points, lines, and figures on a coordinate plane. This branch of mathematics uses coordinates (x, y) to illustrate geometric principles with precision and allows us to execute calculations involving distances, midpoints, and slopes.Key Concepts in Coordinate Geometry

• **Points and Coordinates**: Every point on a plane can be uniquely identified using a pair of coordinates (x, y).
• **Distance Formula**: Used to calculate the distance between two points, given as \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].
• **Midpoint Formula**: Finds the midpoint by \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
• **Line Equations**: Describe lines using equations, typically in slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In the exercise at hand, coordinate geometry aids in analyzing and determining the relationship between the two lines, \( L_1 \) and \( L_2 \), by leveraging their slopes and the coordinates through which they pass. This integration of algebra with geometry serves as a powerful tool in understanding spatial relationships on a plane.