Point of Intersection of Two Lines Formula | Solved Examples & Problems

Introduction

The point of intersection of two lines is a fundamental concept in coordinate geometry. It refers to the exact location where two straight lines cross each other on a Cartesian plane. The intersection point satisfies the equations of both lines simultaneously, making it a crucial tool in solving problems related to geometry, physics, and engineering.

Understanding the Concept

When two lines are drawn on a coordinate plane, they may have different relationships with each other:

1. Intersecting Lines: Two lines cross at a single unique point, which is called the point of intersection.
2. Parallel Lines: Two lines never meet, meaning they have no point of intersection.
3. Coincident Lines: Two lines completely overlap each other, implying they have infinitely many points of intersection.

For intersecting lines, their equations hold true at a particular set of coordinates (x, y), which is found by solving the given equations simultaneously.

Point of Intersection of Two Lines Formula

Point of intersection means the point at which two lines intersect. These two lines are represented by the equation a₁x + b₁y + c₁= 0  and a₂x + b₂y + c₂ = 0, respectively. Given figure illustrate the point of intersection of two lines.

If we consider two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 the point of intersection of these two lines is given by:

Point of Intersection (x, y) = ((b₁×c₂ − b₂×c₁)/(a₁×b₂ − a₂×b₁), (c₁×a₂ − c₂×a₁)/(a₁×b₂ − a₂×b₁))

Fig.1 : Point of intersection

Derivation of the point of intersection of two lines: 

Given equations:

a₁x + b₁y + c₁ = 0 ………..(1)

a₂x + b₂y + c₂ = 0 ………(2)

Solving the equations using cross multiplication method:

       x     y     1

    b₁    c₁    a₁    b₁

    b₂    c₂    a₂    b₂

On cross-multiplying the constants we obtain:

→ x/(b₁*c₂ – b₂* c₁) = y/(c₁*a₂-c₂*a₁) = 1/(a₁*b₂-a₂*b₁)

Solving for x:

→ x/(b1*c2 – b2* c1) = 1/(a1*b2-a2*b1) 

→ x = (b1*c2 – b2* c1)/(a1*b2-a2*b1)

Solving for y:

→ y/(c1*a2-c2*a1) = 1/(a1*b2-a2*b1)

→ y=(c1*a2−c2*a1)/(a1*b2−a2*b1)

Hence point of intersection:

(x,y) = [(b₁×c₂ − b₂×c₁)/(a₁×b₂ − a₂×b₁), (c₁×a₂ − c₂×a₁)/(a₁×b₂ − a₂×b₁)]

If two lines are parallel they never intersect each other:

Condition for two lines a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 to be parallel 

 a₁/b₁ = a₂/b₂. 

Sample Problems on Point of Intersection of Two Lines Formula

Question 1: Find the point of intersection of line 3x + 4y + 5 = 0, 2x + 5y +7 = 0.

Solution:

The point of intersection of two lines is given by :

 (x, y) = ((b1*c2−b2*c1)/(a1*b2−a2*b1), (c1*a2−c2*a1)/(a1*b2−a2*b1))

 a1 = 3, b1 = 4, c1 = 5

 a2 = 2, b2 = 5, c2 = 7

 (x,y) = ((28-25)/(15-8), (10-21)/(15-8))

 (x,y) = (3/7,-11/7)

Question 2: Find the point of intersection of line 9x + 3y + 3 = 0, 4x + 5y + 6 = 0.

Solution:

The point of intersection of two lines is given by :

 (x,y) = ((b1*c2−b2*c1)/(a1*b2−a2*b1), (c1*a2−c2*a1)/(a1*b2−a2*b1))

 a1 = 9, b1 = 3, c1 = 3

 a2 = 4, b2 = 5, c2 = 6

 (x, y) = ((18-15)/(45-15), (54-12)/(45-15))

 (x, y) = (1/10, 7/5)

Question 3: Check if the two lines are parallel or not  2x + 4y + 6 = 0, 4x + 8y + 6 = 0

Solution:

To check whether the lines are parallel or not we need to check a1/b1 = a2/b2

a1 = 2, b1 = 4

a2 = 4, b2 = 8

2/4 = 4/8

1/2 = 1/2

Since the condition is satisfied the lines are parallel and can’t intersect each other.

Question 4: Check if the two lines are parallel or not  3x + 4y + 8 = 0, 4x + 8y + 6 = 0

Solution:

To check whether the lines are parallel or not we need to check a1/b1 = a2/b2

a1 = 3, b1 = 4

a2 = 4, b2 = 8

3/4 is not equal to 4/8

Since the condition is not satisfied the lines are not parallel.

Question 5: Check whether the point (3, 5) is point of intersection of lines 2x + 3y – 21 = 0, x + 2y – 13 = 0.

Solution:

A point to be a point of intersection it should satisfy both the lines.

Substituting (x,y) = (3,5) in both the lines

Check for equation 1: 2*3 + 3*5 – 21 =0 —-> satisfied

Check for equation 2: 3 + 2* 5 -13 =0 —-> satisfied

Since both the equations are satisfied it is a point of intersection of both the lines.

Question 6: Check whether the point (2, 5) is point of intersection of lines x + 3y – 17 = 0, x + y – 13 = 0

Solution:

A point to be a point of intersection it should satisfy both the lines.

Substituting (x,y) = (2,5) in both the lines

Check for equation 1: 2+ 3*5 – 17 =0 —-> satisfied

Check for equation 2: 7 -13 = -6  —>not satisfied

Since both the equations are not satisfied it is not a point of intersection of both the lines.

Question 7: Find the point of intersection of lines x = -2 and 3x + y + 4 = 0

Solution:

On substituting x = -2 in 3x + y + 4 = 0

-6 + y + 4 = 0;

y = 2;

So the point of intersection is (x,y) = (-2,2)                   

Practice Problems on Point of Intersection of Two Lines Formula

1. Find the point of intersection of the lines represented by the equations: 2x + 3y – 6 = 0 and 4x − y + 8 = 0.

2. Determine the point of intersection for the following pair of lines: 5x − y − 4 = 0 and 3x + 2y − 7 = 0.

3. Calculate the intersection point of these lines: x − 2y + 1 = 0 and 2x + y − 5 = 0.

4. Find the point of intersection of the given lines: 3x + 4y − 12 = 0 and 6x − y + 2 = 0.

Applications of Intersection of Two Lines

The concept of the point of intersection is widely used in:

• Solving systems of linear equations in algebra.
• Analyzing motion in physics, such as the paths of moving objects.
• Engineering and architecture, where determining intersecting points is crucial for design.
• Computer graphics and robotics, where paths and object intersections are calculated.

Special Cases in Intersection of Two Lines

• Unique Intersection: If two non-parallel lines intersect, they meet at one unique point.
• No Intersection: If two lines are parallel, they do not meet at any point.
• Infinite Intersections: If two lines are coincident, every point on one line is also a point on the other.

Summary

The point of intersection of two lines plays a crucial role in coordinate geometry and algebra. Understanding this concept helps students solve complex mathematical problems involving line equations and graphical solutions. Mastering this topic is essential for students preparing for CBSE Board Exams and JEE Mains & Advanced.

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