Dividing a Line Segment by the Y-Axis: A Comprehensive Guide

In coordinate geometry, understanding how a line segment is divided by the Y-axis is crucial for a variety of applications, including line segment geometry and problem solving. This comprehensive guide will explore the method to find the ratio in which a line segment joining two points is divided by the Y-axis, using step-by-step calculations and examples.

Introduction to the Problem

Given two points A(2, 3) and B(-2, 1), we want to find the ratio in which the line segment connecting A and B is divided by the Y-axis. The Y-axis is the vertical line defined by x 0.

Methodology

Step 1: Identify the Coordinates of Points

The coordinates of the given points are:

Step 2: Determine the Equation of the Line Segment

The line segment can be expressed parametrically as:

x 2 - 4t (where t varies from 0 to 1)

y 3 - 2t

Here, t 0 corresponds to point A, and t 1 corresponds to point B.

Step 3: Find Where the Line Intersects the Y-Axis

The Y-axis is defined by the equation x 0. Set x to 0 in the equation for x:

2 - 4t 0 implies 4t 2 implies t 1/2

Substitute t 1/2 into the y equation:

y 3 - 2(1/2) 3 - 1 2

Thus, the point of intersection on the Y-axis is (0, 2).

Step 4: Calculate the Distances

The distance from A(2, 3) to (0, 2):
distance_{A} sqrt{(2-0)^2 (3-2)^2} sqrt{2^2 1^2} sqrt{4 1} sqrt{5}

The distance from B(-2, 1) to (0, 2):
distance_{B} sqrt{(-2-0)^2 (1-2)^2} sqrt{(-2)^2 (-1)^2} sqrt{4 1} sqrt{5}

Step 5: Determine the Ratio

Since both distances are equal, the Y-axis divides the segment joining points A and B in the ratio:

ratio distance_{A} : distance_{B} sqrt{5} : sqrt{5} 1 : 1

Additional Insights and Variations

Several methods can be used to solve this problem, including:

Method 1: Using the Section Formula

Let point P on the Y-axis divide the segment in K:1. The coordinates of P are given by:

x -2K/3, y 3K/3

Since the point lies on the Y-axis, x 0:

-2K/3 0 implies K 1

Hence, the ratio is 1:1.

Method 2: Using the Section Formula

Let the Y-axis divide the line segment joining points A(2, 3) and B(-2, 1) in K:1 if the x-coordinate will be 0. Using the section formula:

K(-2) 2 0 and K 1 2K

From K(-2) 2 0, we get K 2/2 1

Therefore, the Y-axis divides the line segment joining points A and B in 1:1 ratio.

Method 3: Equation of the Line

The equation of the line joining A(2, 3) to B(-2, 1) is:

2(y - 3) x - 2

Putting x 0 to find the intersection with the Y-axis, we get:

2(y - 3) -2

y 2, so the point is (0, 2)

Since the point (0, 2) lies between A and B, let the point C(0, 2) divide the line segment in m:n ratio. The equations for m:n are satisfied if m:n 1:1.

Conclusion

In conclusion, the Y-axis divides the line segment joining points A(2, 3) and B(-2, 1) in the ratio 1:1. This proof demonstrates the utility of various methods, including parametric equations, section formula, and the equation of the line, in solving such problems effectively.