Hello,Welcome to StudentBro.
Hello,Welcome to StudentBro.
Hello,Welcome to StudentBro.
The Pair of Straight Lines is an important chapter in JEE Maths, extending the concept of a single line to two lines represented together. Problems involve intersections, angles, concurrent lines, and coordinate geometry applications. Mastering this topic helps students solve complex line-related questions efficiently. At StudentBro.in, we provide a comprehensive guide covering formulas, properties, examples, and practice problems to help JEE aspirants excel in pair of straight lines.
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A pair of straight lines refers to two lines represented by a single second-degree equation:
ax² + 2hxy + by² + 2gx + 2fy + c = 0
If it represents two intersecting straight lines, it can be factored into (L₁)(L₂) = 0
Lines may intersect, be parallel, or coincide
Example: x² - y² = 0 → (x - y)(x + y) = 0, representing lines y = x and y = -x
If the pair of lines is represented by ax² + 2hxy + by² = 0, the slopes (m₁ and m₂) are:
m₁, m₂ = [- (h) ± √(h² - ab)] / b
Used to determine the angle between lines
Example: x² - 3xy + 2y² = 0 → slopes m₁ = 1, m₂ = 2
If slopes of two lines are m₁ and m₂, angle θ between them:
tan θ = |(m₁ - m₂) / (1 + m₁ m₂)|
If θ = 90° → lines are perpendicular
If θ = 0° → lines are parallel
Example: m₁ = 1, m₂ = -1 → tan θ = |(1 + 1)/(1 -1)| → θ = 90°
A second-degree equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of straight lines if:
abc + 2fgh - af² - bg² - ch² = 0
This is used in JEE Advanced problems to check whether an equation represents two lines.
For lines represented by ax² + 2hxy + by² + 2gx + 2fy + c = 0, the point of intersection can be found by solving the simultaneous equations of the two lines after factorization.
Intersection point is also called the concurrent point if multiple lines pass through it
Example: x² - xy - 6y² = 0 → (x - 3y)(x + 2y) = 0 → Lines intersect at origin (0,0)
If the lines are parallel and represented as L₁: y = mx + c₁ and L₂: y = mx + c₂, the distance (d) is:
d = |c₂ - c₁| / √(1 + m²)
Example: Lines y = 2x + 3 and y = 2x - 1 → d = |(-1) - 3| / √(1 + 4) = 4/√5
To find the equations of individual lines from ax² + 2hxy + by² = 0:
Solve for slopes using m₁, m₂ = [-h ± √(h² - ab)] / b
Use point of intersection (often origin) to write line equations as y = m₁x and y = m₂x
Example: x² - y² = 0 → slopes m₁ = 1, m₂ = -1 → Lines: y = x, y = -x
Solve intersection, angles, and concurrent lines problems
Determine parallel or perpendicular lines
Factorize second-degree equations to find individual lines
Solve coordinate geometry questions in JEE Main & Advanced
Used in mechanics, geometry, and linear algebra applications
Memorize slope formulas and angle between lines formula
Learn conditions for pair of lines using abc + 2fgh - af² - bg² - ch² = 0
Factorize homogeneous second-degree equations for quick solution
Use parallel and perpendicular line conditions to simplify problems
Practice previous year JEE questions for speed and accuracy
At StudentBro.in, students can access:
Step-by-step explanations for equations, slopes, intersections, angles, and concurrent lines
Worked examples for parallel, perpendicular, and intersecting lines
Practice questions with solutions and shortcuts
Tips for quick problem-solving and exam strategy
Revision notes for last-minute preparation
Find slopes of lines represented by x² - 3xy + 2y² = 0
Determine the angle between lines x² - y² = 0
Find point of intersection of lines x² - xy - 6y² = 0
Check if lines 2x - y + 3 = 0 and 4x - 2y - 5 = 0 are parallel
Reduce x² - 5xy + 6y² = 0 into two individual line equations
Regular practice improves accuracy, speed, and confidence in solving pair of straight lines problems in JEE exams.
Pair of Straight Lines is a critical chapter in JEE Maths. Understanding slope, angle, intersection, factorization, and concurrent lines allows students to solve complex coordinate geometry problems efficiently. At StudentBro.in, we provide a complete guide from basics to advanced problem-solving, making pair of straight lines simple, practical, and exam-oriented.
Mastering this topic ensures students can tackle intersection, parallelism, and angular problems confidently in both JEE Main and Advanced exams.
