Finding the Ratio in Which the Y-Axis Divides a Line Segment
Understanding the division of a line segment by the y-axis can be crucial in coordinate geometry. In this article, we'll explore the concept through an example and apply the section formula to find the required ratio.
Introduction
Given two points, we can determine the ratio in which the y-axis divides the line segment joining them using the section formula. The y-axis is defined by the equation x 0. We will explore this method using specific coordinates for the two points.
Problem Statement
The problem involves finding the ratio in which the y-axis divides the line segment joining the points -3, -4 and 1, -2.
Section Formula
The section formula for a point dividing a line segment in the ratio m:n is given by:
Px 2 - nx1}{m n}
Py 2 - ny1}{m n}
Here, (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment, and P is the point dividing the segment in the ratio m:n.
Solving for the Ratio
Let's denote the unknown ratio as k:1. We will find the coordinates of the point P on the y-axis (i.e., with coordinates (0, y)) that divides the segment in this ratio.
Substituting the coordinates of the endpoints (-3, -4) and (1, -2), and the ratio k:1 into the section formula, we get:
Px
Since P lies on the y-axis, Px 0. Therefore, we set up the equation:
0
Setting the numerator equal to zero gives:
k - 3 0
Solving for k, we find:
k 3
Thus, the ratio in which the y-axis divides the line segment joining the points -3, -4 and 1, -2 is:
boxed{3:1}
Verification
To verify, we can find the point of intersection of the line segment with the y-axis. The line joining the points -3, -4 and 1, -2 can be expressed by the equation:
2y - x - 5 0
Setting x 0 (the y-axis), we get:
2y - 5 0
y -
So, the point of intersection with the y-axis is (0, -).
Conclusion
In summary, the y-axis divides the line segment joining the points -3, -4 and 1, -2 in the ratio 3:1. This can be verified by both the section formula and the point of intersection method.