Understanding Perpendicular Lines and Their Slopes

When two lines in a coordinate plane are perpendicular, their slopes have a special relationship. The product of the slopes of two perpendicular lines is always -1, provided neither line is vertical or horizontal. Vertical lines have undefined slopes, and horizontal lines have a slope of 0. A vertical line is perpendicular to a horizontal line.

Let's consider the given equations of the lines:

• Line 1: \(y + px = 1\)
• Line 2: \(y - qx = 2\)

To find the slopes of these lines, we need to rewrite them in the slope-intercept form, which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Converting Equations to Slope-Intercept Form

For the first line, \(y + px = 1\), we isolate \(y\) by subtracting \(px\) from both sides:

\(\qquad y = -px + 1\)

Comparing this to \(y = mx + c\), the slope of the first line, let's call it \(m_1\), is:

\(\qquad m_1 = -p\)

For the second line, \(y - qx = 2\), we isolate \(y\) by adding \(qx\) to both sides:

\(\qquad y = qx + 2\)

Comparing this to \(y = mx + c\), the slope of the second line, let's call it \(m_2\), is:

\(\qquad m_2 = q\)

Applying the Condition for Perpendicular Lines

The problem states that the two lines are perpendicular. The condition for two non-vertical perpendicular lines is that the product of their slopes is -1. That is, \(m_1 \times m_2 = -1\).

Substitute the slopes we found (\(m_1 = -p\) and \(m_2 = q\)) into this condition:

\(\qquad (-p) \times (q) = -1\)

\(\qquad -pq = -1\)

To find the relationship between \(p\) and \(q\), we can multiply both sides of the equation by -1:

\(\qquad pq = 1\)

Alternatively, we can move the constant term to the left side by subtracting 1 from both sides:

\(\qquad pq - 1 = 0\)

Comparing with the Given Options

Let's compare our derived relationship, \(pq - 1 = 0\), with the provided options:

• Option 1: \(pq + 1 = 0\)
• Option 2: \(p + q + 1 = 0\)
• Option 3: \(pq - 1 = 0\)
• Option 4: \(p - q + 1 = 0\)

Our derived condition \(pq - 1 = 0\) matches Option 3.

Summary of Steps

Revision Table: Perpendicular Lines

Additional Information on Line Relationships

Besides perpendicular lines, lines can also be parallel or intersecting non-perpendicularly.

• Parallel Lines: Two lines are parallel if they have the same slope and different y-intercepts. The condition is \(m_1 = m_2\). If they have the same slope and the same y-intercept, they are the same line (coincident).
• Intersecting Lines: Lines that cross each other at a single point. If their slopes are different (\(m_1 \neq m_2\)), they intersect. Perpendicular lines are a special case of intersecting lines.

Understanding the slopes of lines is fundamental in coordinate geometry for determining the relationship between lines.