Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-32: Find a general form of an equation of the line through the point \(A\) that satisfies the given condition. $$ A(5,-2) $$ (a) parallel to the \(y\)-axis (b) perpendicular to the \(y\)-axis

Step-by-Step Solution

Answer

(a) \( x = 5 \); (b) \( y = -2 \).

Step 1: Understand the Condition (a) Parallel to y-axis

When a line is parallel to the y-axis, it means that the line is a vertical line. A vertical line has an undefined slope, and its equation is of the form \( x = c \), where \( c \) is the x-coordinate of any point on the line.

Step 2: Apply Condition to Point A (a)

Given the point \( A(5, -2) \), the x-coordinate is 5. Therefore, the equation of the line parallel to the y-axis through point A is \( x = 5 \).

Step 3: Understand the Condition (b) Perpendicular to y-axis

When a line is perpendicular to the y-axis, it is a horizontal line. A horizontal line has a slope of 0, and its equation is of the form \( y = c \), where \( c \) is the y-coordinate of any point on the line.

Step 4: Apply Condition to Point A (b)

Given the point \( A(5, -2) \), the y-coordinate is -2. Therefore, the equation of the line perpendicular to the y-axis through point A is \( y = -2 \).

Key Concepts

Parallel Lines

When we talk about parallel lines in coordinate geometry, we refer to lines that never meet, regardless of how far they are extended. This property is due to their having the same slope. In simpler terms, parallel lines will always maintain the same distance from each other.

Here’s how you can identify parallel lines on a graph:

• Lines with equal slopes are parallel (except vertical lines, which have undefined slopes but can still be parallel).
• The equation of a line parallel to another can be obtained by keeping the slope constant and adjusting the y-intercept.

When a line is parallel to the y-axis, as mentioned in the exercise, it becomes a vertical line. These lines don’t have a slope, so we describe them using the equation of the form \( x = c \) where \( c \) is a constant, like the x-coordinate of point A in our example.

Perpendicular Lines

Perpendicular lines are those that intersect at a right angle (90 degrees). This perfect crossing leads to an interesting relationship between their slopes.

Here's a quick rundown of how to determine if two lines are perpendicular:

• If one line has a slope \( m \), the line perpendicular to it will have a slope of \(-\frac{1}{m}\).
• If one of the lines is horizontal (slope of 0), then the perpendicular line will be vertical (undefined slope), and vice versa.

In the exercise, when a line is perpendicular to the y-axis, it forms a horizontal line. This line's slope is 0, and its equation takes the form of \( y = c \), where \( c \) is the y-coordinate of any point the line passes through.

Coordinate Geometry

Coordinate geometry is a branch of geometry where points are placed in a plane using coordinate systems. This approach provides a connection between algebra and geometry through graphs of shapes and equations describing their dimensions.

In solving exercises, we often rely on:

• The coordinate plane, which is structured by the x-axis (horizontal) and y-axis (vertical).
• Points represented as \((x, y)\), showing their position relative to both axes.
• Lines, defined by equations relating x and y, such as \( y = mx + b \).

With these tools, we use coordinate geometry to solve problems involving points, lines, and shapes, enhancing our understanding of spatial relationships and algebraic representations.

Slope

Slope is a measure of how steep a line is. It tells us how much the line rises or falls as it moves horizontally. The slope is a vital concept in understanding linear equations and graphing.

Here's how you can explore slope further:

• Calculated as "rise over run", meaning the change in y divided by the change in x.
• A positive slope indicates the line ascends, while a negative slope means it descends.
• Zero slope results in a horizontal line, and an undefined slope corresponds to a vertical line.

Knowing the slope helps us determine the direction and steepness of lines, find equations from points and slopes, and understand the geometric properties of parallel and perpendicular lines.