JEE Maths Notes: Chapter on Pair of Straight Lines

1. Introduction to Pair of Straight Lines

A pair of straight lines represents two lines that can either intersect, be parallel, or coincide with each other. The concept of pair of straight lines is crucial in coordinate geometry. These lines can be represented algebraically by quadratic equations and have significant applications in solving problems related to geometry, algebra, and calculus. Understanding the relationship between the lines in a pair helps in analyzing the geometric configuration and solving intersection or parallelism problems.

2. Equation of Pair of Straight Lines

The general equation of a pair of straight lines passing through the origin is represented as:

Ax² + 2Hxy + By² = 0

Here, A, H, and B are constants. The discriminant of this equation plays a crucial role in determining the nature of the lines. If H² - AB = 0, the equation represents two coincident lines, and if H² - AB > 0, it represents a pair of real and distinct lines. If H² - AB < 0, the equation represents imaginary lines.

3. Conditions for a Pair of Lines

For the equation Ax² + 2Hxy + By² = 0 to represent a pair of straight lines, certain conditions must be satisfied:

• Condition 1: The equation must be homogeneous, i.e., it must not have any constant term.
• Condition 2: The discriminant H² - AB determines the nature of the lines. It must be greater than or equal to zero for real lines.
• Condition 3: The lines can be parallel, coincident, or intersecting based on the values of the constants A, B, and H.

These conditions help in distinguishing different types of line pairs and analyzing their geometric properties.

4. Pair of Lines with a Common Vertex

When two lines intersect at the origin (i.e., they pass through a common vertex), their equation is of the form:

Ax² + 2Hxy + By² + 2Gx + 2Fy = 0

This equation represents a pair of lines that have a common point of intersection, typically the origin. To find the equation of the pair of lines, we need to factorize the quadratic equation and identify the individual line equations. The constants G and F are related to the linear terms in the equation.

5. Homogeneous Equation of a Pair of Lines

A homogeneous equation of a pair of straight lines through the origin is given by:

Ax² + 2Hxy + By² = 0

The solution of this equation yields two straight lines that pass through the origin and are represented by two distinct linear factors. The relationship between the coefficients A, B, and H determines whether the lines are real, parallel, coincident, or imaginary.

6. Angle Between a Pair of Straight Lines

The angle θ between the two lines represented by the equation Ax² + 2Hxy + By² = 0 can be found using the formula:

tan(θ) = ±√[(m₁ - m₂)² / (1 + m₁m₂)]

Where m₁ and m₂ are the slopes of the two lines. This formula is used to find the angle between the two lines when the slopes are known. The formula is derived from the concept of the dot product in vector geometry.

7. Condition for Perpendicular and Parallel Lines

For the pair of lines represented by the equation Ax² + 2Hxy + By² = 0, the condition for the lines to be:

• Perpendicular: The condition is H² = AB.
• Parallel: The condition is H² = AB with A = B and H = 0.

These conditions are derived from the geometric properties of lines and provide insight into the relationship between the lines in the pair.

8. Pair of Lines with the Same Slope

When both lines of the pair have the same slope, the equation can be written as:

(x - x₁)(x - x₂) = 0

This equation represents a pair of straight lines with the same slope, which are either coincident or parallel, depending on the relative positions of x₁ and x₂.

9. Equation of Pair of Lines in Terms of Slopes

If the slopes of the two lines are m₁ and m₂, the equation of the pair of lines can be written as:

y = m₁x + c₁
    y = m₂x + c₂

Where m₁ and m₂ are the slopes of the lines and c₁ and c₂ are the y-intercepts. These equations help in representing the lines geometrically and algebraically for further analysis.

10. Applications of Pair of Straight Lines

The concept of pair of straight lines has several applications in mathematics, physics, and engineering. Some of the key applications include:

• Geometric Interpretation: Analyzing the angles between lines, finding perpendicular bisectors, and understanding the behavior of lines in different configurations.
• Coordinate Geometry: Solving problems involving distances between lines, angle between lines, and finding equations of lines passing through given points.
• Physics: Modeling motion in two dimensions, analyzing forces and velocities, and solving vector-based problems.
• Engineering: Designing structures, understanding stress and strain along lines, and solving problems related to materials science.

11. Practice Problems and Examples

This section provides practice problems to enhance understanding and problem-solving skills related to the pair of straight lines. The problems involve finding the equations of lines, calculating angles, determining conditions for parallelism and perpendicularity, and analyzing intersections. Solutions and step-by-step explanations are provided to reinforce the concepts.

12. Summary and Key Takeaways

In summary, the chapter on Pair of Straight Lines focuses on understanding the geometric and algebraic properties of two straight lines. It covers the equation of a pair of lines, conditions for parallelism, perpendicularity, and angle calculation, as well as the applications of these concepts. Mastery of these topics is essential for solving a wide range of problems in coordinate geometry and preparing for competitive exams like JEE.