Edexcel IAL - Pure Maths 1- 2.2 Conditions for Parallel and Perpendicular Lines- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 2.2 Conditions for Parallel and Perpendicular Lines -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 2.2 Conditions for Parallel and Perpendicular Lines -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- Conditions for Parallel and Perpendicular Lines
Conditions for Two Straight Lines to Be Parallel or Perpendicular
Two lines in the plane can be compared using their slopes. If the slopes satisfy specific relationships, the lines will be parallel or perpendicular.
General Line Form:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)
The slope of a line in general form is:
\( m = -\dfrac{a}{b} \) (provided \( b \neq 0 \))
Conditions
| Relationship | Condition |
Parallel lines | \( m_1 = m_2 \) or \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \) |
Perpendicular lines | \( m_1 \cdot m_2 = -1 \) or \( a_1a_2 + b_1b_2 = 0 \) |
Interpretation of the Conditions
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals.
- In vector form, the direction vectors are proportional for parallel lines.
- For perpendicular lines, the product \( a_1a_2 + b_1b_2 = 0 \) indicates their direction vectors are perpendicular.
Example
Determine whether the lines \( y = 3x + 1 \) and \( y = 3x – 5 \) are parallel, perpendicular or neither.
▶️ Answer / Explanation
Slope of first line: \( m_1 = 3 \)
Slope of second line: \( m_2 = 3 \)
Since \( m_1 = m_2 \), the lines are parallel.
Example
Determine the relationship between the lines \( 2x – 3y + 4 = 0 \) and \( 3x + 2y – 7 = 0 \).
▶️ Answer / Explanation
Slope of first line:
\( 2x – 3y + 4 = 0 \Rightarrow -3y = -2x – 4 \Rightarrow y = \dfrac{2}{3}x + \dfrac{4}{3} \)
So \( m_1 = \dfrac{2}{3} \)
Slope of second line:
\( 3x + 2y – 7 = 0 \Rightarrow 2y = -3x + 7 \Rightarrow y = -\dfrac{3}{2}x + \dfrac{7}{2} \)
So \( m_2 = -\dfrac{3}{2} \)
Compute product:
\( m_1 m_2 = \dfrac{2}{3} \cdot \left(-\dfrac{3}{2}\right) = -1 \)
Since \( m_1 m_2 = -1 \), the lines are perpendicular.
Example
Show whether the lines \( 4x + 7y – 9 = 0 \) and \( 8x + 14y + 3 = 0 \) are parallel, perpendicular or neither.
▶️ Answer / Explanation
Compare coefficients:
\( \dfrac{4}{8} = \dfrac{1}{2} \)
\( \dfrac{7}{14} = \dfrac{1}{2} \)
Since:
\( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \)
The lines are parallel.
The constant term comparison does not matter; only slopes determine parallelism.