Mastering Family of Straight Lines for Competitive Exams

Welcome to this module on the 'Family of Straight Lines'! This concept is a powerful tool in coordinate geometry, allowing us to represent a collection of lines that share a common property, often a point of intersection. Understanding families of lines is crucial for solving various problems in competitive exams like JEE Mathematics, particularly in calculus and algebra applications.

What is a Family of Straight Lines?

A family of straight lines is a set of lines that satisfy a particular condition. Instead of defining a single line, we define a general equation that represents multiple lines. The most common types of families are based on a common point of intersection.

Family of Lines Passing Through a Fixed Point

Consider two intersecting lines: $L_1: a_1x + b_1y + c_1 = 0$ and $L_2: a_2x + b_2y + c_2 = 0$ . The family of lines passing through their point of intersection can be represented by the equation $L_1 + \lambda L_2 = 0$ , where $\lambda$ is a non-zero arbitrary constant. This equation, $(a_1x + b_1y + c_1) + \lambda(a_2x + b_2y + c_2) = 0$ , represents any line passing through the intersection of $L_1$ and $L_2$ (except $L_2$ itself).

The equation $L_1 + \lambda L_2 = 0$ is the cornerstone of families of lines passing through the intersection of $L_1$ and $L_2$.

This form is incredibly useful because it encapsulates an infinite number of lines with a single parameter ( $\lambda$ ). By varying $\lambda$ , we can obtain any line passing through the common point.

To understand why $L_1 + \lambda L_2 = 0$ works, let $(x_0, y_0)$ be the point of intersection of $L_1$ and $L_2$ . This means $a_1x_0 + b_1y_0 + c_1 = 0$ and $a_2x_0 + b_2y_0 + c_2 = 0$ . Substituting $(x_0, y_0)$ into the family equation gives $(0) + \lambda(0) = 0$ , which is always true. Thus, any line in this family passes through $(x_0, y_0)$ . To get a specific line, we need an additional condition, such as passing through another point, having a specific slope, or being parallel/perpendicular to another line.

What is the general equation for a family of lines passing through the intersection of

L_1 = 0

and

L_2 = 0

$L_1 + \lambda L_2 = 0$ , where $\lambda$ is a non-zero constant.

Alternative Form: Family of Lines Through a Point $(x_1, y_1)$

A line passing through a fixed point $(x_1, y_1)$ can be represented in various forms. The slope-intercept form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope. This can be rewritten as $y - y_1 - m(x - x_1) = 0$ . This equation represents a family of lines passing through $(x_1, y_1)$ , with $m$ as the parameter. A special case is the vertical line $x = x_1$ , which cannot be represented by this form but passes through $(x_1, y_1)$ .

Remember that the form $y - y_1 = m(x - x_1)$ covers all lines through $(x_1, y_1)$ except the vertical line $x = x_1$ .

Other Types of Families

Beyond lines intersecting at a point, families can be defined by other properties:

Family of parallel lines: $ax + by + k = 0$ , where $k$ is the parameter. All these lines have the same slope $-a/b$ .
Family of perpendicular lines: $bx - ay + k = 0$ , where $k$ is the parameter. These lines have a slope $b/a$ , which is the negative reciprocal of $-a/b$ .

Family Type	General Equation	Key Property
Through intersection of $L_1=0, L_2=0$	$L_1 + \lambda L_2 = 0$	Passes through a fixed point (intersection of $L_1, L_2$ )
Through point $(x_1, y_1)$	$y - y_1 = m(x - x_1)$	Passes through a fixed point $(x_1, y_1)$
Parallel to $ax+by+c=0$	$ax + by + k = 0$	Same slope
Perpendicular to $ax+by+c=0$	$bx - ay + k = 0$	Slope is negative reciprocal

Applications in Problem Solving

Family of lines simplifies problems where multiple lines are involved. For instance, finding a line that passes through the intersection of two given lines and also satisfies another condition (like passing through a third point or being tangent to a circle) becomes straightforward using the $L_1 + \lambda L_2 = 0$ form. You simply substitute the coordinates of the third point or use the tangency condition to find the value of $\lambda$ .

Imagine two roads, Road A and Road B, that cross at a specific intersection. The 'family of lines' concept is like having a map that shows all possible paths you could take starting from that intersection. Each path is a different line, and the parameter $\lambda$ is like a dial that lets you select which path to take. For example, if Road A is $x+y-2=0$ and Road B is $x-y=0$ , the family of lines passing through their intersection is $(x+y-2) + \lambda(x-y) = 0$ . If you want a line that also passes through the origin (0,0), you'd substitute x=0, y=0 into the family equation: $(0+0-2) + \lambda(0-0) = 0$ , which gives $-2 = 0$ , an impossibility. This means no line in this family passes through the origin. Let's try a point (3,1). Substituting: $(3+1-2) + \lambda(3-1) = 0 \implies 2 + 2\lambda = 0 \implies \lambda = -1$ . The line is $(x+y-2) - (x-y) = 0 \implies 2y - 2 = 0 \implies y=1$ . This line passes through the intersection of $x+y-2=0$ and $x-y=0$ (which is (1,1)) and also through (3,1).

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Text-based content

Library pages focus on text content

What is the primary advantage of using the family of lines concept in problem-solving?

It simplifies problems involving multiple lines by representing them with a single equation and a parameter, making it easier to apply additional conditions.

Key Takeaways for Exams

When approaching problems involving families of lines:

Identify the type of family: Is it through a point, or through the intersection of two lines?
Write the general equation: Use the appropriate form ( $L_1 + \lambda L_2 = 0$ or $y - y_1 = m(x - x_1)$ ).
Use the given condition: Substitute coordinates or apply slope/intercept conditions to find the parameter ( $\lambda$ or $m$ ).
Simplify the final equation: Ensure the line equation is in its simplest form.

Learning Resources

Family of Straight Lines - Concepts and Problems(documentation)

Provides a clear explanation of different types of families of straight lines and includes solved examples.

Family of Straight Lines | JEE Mathematics(blog)

Explains the concept of family of straight lines with a focus on JEE preparation, including formulas and problem-solving techniques.

Straight Lines - Family of Lines(blog)

A concise explanation of the family of lines concept, often found in competitive exam preparation materials.

JEE Math: Straight Lines - Family of Lines(video)

A video tutorial explaining the concept of family of lines with examples relevant to JEE mathematics.

Coordinate Geometry: Straight Lines(documentation)

While broader, this section on lines in LibreTexts often covers foundational concepts that can be extended to families of lines.

NCERT Mathematics Class 11 - Chapter 10: Straight Lines(documentation)

The official NCERT textbook provides a fundamental understanding of straight lines, which is essential for grasping families of lines.

Khan Academy: Lines and their properties(tutorial)

Offers comprehensive tutorials on various aspects of lines, including slope, intercepts, and equations, which are building blocks for family of lines.

Understanding Families of Lines in Coordinate Geometry(blog)

This resource breaks down the concept of families of lines, offering clear definitions and examples for students.

JEE Advanced Mathematics - Straight Lines(wikipedia)

Math Stack Exchange is a great place to find discussions and solutions to specific problems related to straight lines and families of lines.

The Concept of Locus in Coordinate Geometry(blog)

Understanding locus is often intertwined with families of lines, as the family itself can be seen as a locus of lines satisfying certain conditions.